Set familiesFrom Wikipedia, the free encyclopedia

Contents

1 Abstract simplicial complex 11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Almost disjoint sets 52.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Other meanings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Antimatroid 73.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Paths and basic words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Convex geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Join-distributive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.6 Supersolvable antimatroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 Join operation and convex dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.8 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Block design 144.1 Definition of a BIBD (or 2-design) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Symmetric BIBDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.1 Projective planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.2 Biplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.3 Hadamard 2-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Resolvable 2-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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4.4 Generalization: t-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.1 Derived and extendable t-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.5 Steiner systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6 Partially balanced designs (PBIBDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.6.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.6.3 Two associate class PBIBDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Carathéodory’s theorem (convex hull) 225.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Clique complex 256.1 Independence complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Flag complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3 Conformal hypergraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4 Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Combinatorial design 287.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.3 Fundamental combinatorial designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.4 A wide assortment of other combinatorial designs . . . . . . . . . . . . . . . . . . . . . . . . . . 307.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Configuration (geometry) 378.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.3 Duality of configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.4 The number of (n3) configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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8.5 Constructions of symmetric configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.6 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9 Content (measure theory) 429.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.2 Integration of bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.3 Duals of spaces of bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4 Construction of a measure from a content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.4.1 Definition on open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4.2 Definition on all sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4.3 Construction of a measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Dedekind number 4510.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.3 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.4 Summation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.5 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

11 Delta set 4911.1 Definition and related data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.2 Related functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

12 Delta-ring 5312.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

13 Dendroidal set 5413.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

14 Discrete differential geometry 5514.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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15 Disjoint sets 5615.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5815.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5815.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5815.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

16 Dold–Kan correspondence 6016.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

17 Dynkin system 6117.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2 Dynkin’s π-λ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

18 Erdős–Ko–Rado theorem 6318.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.2 Families of maximum size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.3 Maximal intersecting families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

19 Family of sets 6619.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

20 Field of sets 6820.1 Fields of sets in the representation theory of Boolean algebras . . . . . . . . . . . . . . . . . . . . 68

20.1.1 Stone representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.1.2 Separative and compact fields of sets: towards Stone duality . . . . . . . . . . . . . . . . . 68

20.2 Fields of sets with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.2.1 Sigma algebras and measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.2.2 Topological fields of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.2.3 Preorder fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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20.2.4 Complex algebras and fields of sets on relational structures . . . . . . . . . . . . . . . . . . 7020.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7120.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7120.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

21 Finite character 7221.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

22 Finite intersection property 7322.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7322.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7322.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7322.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7422.5 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7422.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7422.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

23 Fisher’s inequality 7523.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

24 Generalized quadrangle 7824.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.5 Generalized quadrangles with lines of size 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8024.6 Classical generalized quadrangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8024.7 Non-classical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8124.8 Restrictions on parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8124.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

25 Greedoid 8225.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8225.2 Classes of greedoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8225.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.4 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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26 Helly family 8526.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.2 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.3 Helly dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.4 The Helly property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

27 Helly’s theorem 8827.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8827.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8827.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

28 Incidence structure 9128.1 Formal definition and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9128.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9128.3 Dual structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.4 Other terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

28.4.1 Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.4.2 Block designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

28.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.5.1 Incidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.5.2 Pictorial representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.5.3 Incidence graph (Levi graph) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

28.6 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

29 Kan fibration 9929.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9929.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10229.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10229.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10229.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

30 Kirkman’s schoolgirl problem 10330.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10330.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10430.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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30.4 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10430.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10430.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

31 Kruskal–Katona theorem 10831.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

31.1.1 Statement for simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10831.1.2 Statement for uniform hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

31.2 Ingredients of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10831.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

32 Levi graph 11032.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11032.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11132.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

33 Matroid 11233.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

33.1.1 Independent sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11233.1.2 Bases and circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11233.1.3 Rank functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11333.1.4 Closure operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11333.1.5 Flats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11433.1.6 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

33.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11433.2.1 Uniform matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11433.2.2 Matroids from linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11533.2.3 Matroids from graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11733.2.4 Matroids from field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

33.3 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.3.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.3.2 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.3.3 Sums and unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

33.4 Additional terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11933.5 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

33.5.1 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11933.5.2 Matroid partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12033.5.3 Matroid intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12033.5.4 Matroid software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

33.6 Polynomial invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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33.6.1 Characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12033.6.2 Tutte polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

33.7 Infinite matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12233.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12233.9 Researchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12333.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12333.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12333.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12533.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

34 Maximum coverage problem 12734.1 ILP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12734.2 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12734.3 Known extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12734.4 Weighted version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12734.5 Budgeted maximum coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12834.6 Generalized maximum coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

34.6.1 Generalized maximum coverage algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 12834.7 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12934.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12934.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12934.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

35 Monotone class theorem 13035.1 Definition of a monotone class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13035.2 Monotone class theorem for sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

35.2.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13035.3 Monotone class theorem for functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

35.3.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13035.3.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

35.4 Results and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13135.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13135.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

36 Near polygon 13236.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13336.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13336.3 Regular near polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13336.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13436.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13436.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

37 Nerve (category theory) 135

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37.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13537.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13537.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

37.3.1 Most spaces are classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.3.2 The nerve of an open covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.3.3 A moduli example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

37.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

38 Nerve of a covering 13938.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13938.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

39 Noncrossing partition 14039.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14039.2 Lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14039.3 Role in free probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14039.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

40 Partition of a set 14440.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14540.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14540.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14540.4 Refinement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14640.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14640.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14640.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14740.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14740.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

41 Partition regularity 15241.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15241.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

42 Pi system 15442.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15442.2 Relationship to λ-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

42.2.1 The π-λ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15542.3 π-Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

42.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15642.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

42.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15742.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15742.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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43 Polar space 15843.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15843.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15843.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

44 Pro-simplicial set 15944.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

45 Property B 16045.1 Values of m(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16045.2 Asymptotics of m(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16045.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

46 Radon’s theorem 16246.1 Proof and construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16246.2 Topological Radon theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16346.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16346.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16346.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16446.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

47 Ring of sets 16647.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16647.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16747.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16747.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

48 Sauer–Shelah lemma 16848.1 Definitions and statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16948.2 The number of shattered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16948.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16948.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16948.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

49 Segal space 17149.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17149.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

50 Set cover problem 17250.1 Integer linear program formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17250.2 Hitting set formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17250.3 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17250.4 Low-frequency systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17350.5 Inapproximability results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

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50.6 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17350.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17450.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17450.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

51 Shapley–Folkman lemma 17651.1 Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17751.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

51.2.1 Real vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17751.2.2 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17851.2.3 Convex hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17951.2.4 Minkowski addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17951.2.5 Convex hulls of Minkowski sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

51.3 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18151.3.1 Lemma of Shapley and Folkman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18151.3.2 Shapley–Folkman theorem and Starr’s corollary . . . . . . . . . . . . . . . . . . . . . . . 18351.3.3 Proofs and computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

51.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18451.4.1 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18551.4.2 Mathematical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18951.4.3 Probability and measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

51.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19151.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19551.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

52 Sigma-algebra 19752.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

52.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19752.1.2 Limits of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19852.1.3 Sub σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

52.2 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19952.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19952.2.2 Dynkin’s π-λ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19952.2.3 Combining σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19952.2.4 σ-algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20052.2.5 Relation to σ-ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20052.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

52.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20152.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20152.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

52.4 σ-algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20152.4.1 σ-algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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52.4.2 σ-algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20152.4.3 Borel and Lebesgue σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20252.4.4 Product σ-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20252.4.5 σ-algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20252.4.6 σ-algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 20352.4.7 σ-algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 203

52.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20352.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20452.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

53 Sigma-ideal 20553.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

54 Sigma-ring 20654.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.3 Similar concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20754.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

55 Simplex category 20855.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20855.2 Augmented simplex category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20855.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20855.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20955.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

56 Simplicial approximation theorem 21056.1 Formal statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21056.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

57 Simplicial complex 21157.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21257.2 Closure, star, and link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21257.3 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21257.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21357.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21357.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21357.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

58 Simplicial group 21558.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21558.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

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59 Simplicial homotopy 21659.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21659.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

60 Simplicial manifold 21760.1 A manifold made out of simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21760.2 A simplicial object built from manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

61 Simplicial presheaf 21861.1 Homotopy sheaves of a simplicial presheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21861.2 Model structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21861.3 Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21961.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21961.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21961.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21961.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21961.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

62 Simplicial set 22062.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22062.2 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22062.3 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22162.4 Face and degeneracy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22162.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22162.6 The standard n-simplex and the category of simplices . . . . . . . . . . . . . . . . . . . . . . . . . 22262.7 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22262.8 Singular set for a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22362.9 Homotopy theory of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22362.10Simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22362.11History and uses of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22462.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22462.13Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22462.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

63 Sperner’s theorem 22663.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22663.2 Partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22663.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22663.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

63.4.1 No long chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22763.4.2 p-compositions of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22763.4.3 No long chains in p-compositions of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . 22763.4.4 Projective geometry analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

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63.4.5 No long chains in p-compositions of a projective space . . . . . . . . . . . . . . . . . . . . 22863.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22863.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

64 Steiner system 23064.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

64.1.1 Finite projective planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23164.1.2 Finite affine planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

64.2 Classical Steiner systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23164.2.1 Steiner triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23164.2.2 Steiner quadruple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23164.2.3 Steiner quintuple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

64.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23264.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23264.5 Mathieu groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23264.6 The Steiner system S(5, 6, 12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

64.6.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23364.7 The Steiner system S(5, 8, 24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

64.7.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23464.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23464.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23564.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23564.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

65 Symmetric spectrum 23765.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

66 Teichmüller–Tukey lemma 23866.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23866.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23866.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23866.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

67 Tverberg’s theorem 23967.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24067.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24067.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

68 Two-graph 24168.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24168.2 Switching and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24168.3 Adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24368.4 Equiangular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

CONTENTS xv

68.5 Strongly regular graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24368.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24368.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

69 Ultrafilter 24569.1 Formal definition for ultrafilter on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24569.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24669.3 Generalization to partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24669.4 Special case: Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24669.5 Types and existence of ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24669.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24769.7 Ordering on ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24869.8 Ultrafilters on ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24869.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24869.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24869.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

70 Union-closed sets conjecture 25070.1 Equivalent forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25070.2 Families known to satisfy the conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25070.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25170.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25170.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25170.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

71 Universal set 25271.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

71.1.1 Russell’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25271.1.2 Cantor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

71.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25271.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25371.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

71.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25371.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25371.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

72 Universe (mathematics) 25572.1 In a specific context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25572.2 In ordinary mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25572.3 In set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25672.4 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25772.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25872.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

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72.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25872.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

73 Vietoris–Rips complex 25973.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26073.2 Relation to Čech complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26073.3 Relation to unit disk graphs and clique complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 26073.4 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26073.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26073.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26173.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

74 ∞-groupoid 26274.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26274.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26274.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 263

74.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26374.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26774.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

Chapter 1

Abstract simplicial complex

In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of asimplicial complex, consisting of a family of non-empty finite sets closed under the operation of taking non-emptysubsets.[1] In the context of matroids and greedoids, abstract simplicial complexes are also called independencesystems.[2]

1.1 Definitions

A family Δ of non-empty finite subsets of a universal set S is an abstract simplicial complex if, for every set X inΔ, and every non-empty subset Y ⊂ X, Y also belongs to Δ.The finite sets that belong to Δ are called faces of the complex, and a face Y is said to belong to another face X if Y ⊂X, so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complexΔ is itself a face of Δ. The vertex set of Δ is defined as V(Δ) = ∪Δ, the union of all faces of Δ. The elements of thevertex set are called the vertices of the complex. So for every vertex v of Δ, the set v is a face of the complex.The maximal faces of Δ (i.e., faces that are not subsets of any other faces) are called facets of the complex. Thedimension of a face X in Δ is defined as dim(X) = |X| − 1: faces consisting of a single element are zero-dimensional,faces consisting of two elements are one-dimensional, etc. The dimension of the complex dim(Δ) is defined as thelargest dimension of any of its faces, or infinity if there is no finite bound on the dimension of the faces.The complex Δ is said to be finite if it has finitely many faces, or equivalently if its vertex set is finite. Also, Δ issaid to be pure if it is finite-dimensional (but not necessarily finite) and every facet has the same dimension. In otherwords, Δ is pure if dim(Δ) is finite and every face is contained in a facet of dimension dim(Δ).One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertexset of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspondto undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that donot have any incident edges.A subcomplex of Δ is a simplicial complex L such that every face of L belongs to Δ; that is, L ⊂ Δ and L is asimplicial complex. A subcomplex that consists of all of the subsets of a single face of Δ is often called a simplexof Δ. (However, some authors use the term “simplex” for a face or, rather ambiguously, for both a face and thesubcomplex associated with a face, by analogy with the non-abstract (geometric) simplicial complex terminology. Toavoid ambiguity, we do not use in this article the term “simplex” for a face in the context of abstract complexes.)The d-skeleton of Δ is the subcomplex of Δ consisting of all of the faces of Δ that have dimension at most d. Inparticular, the 1-skeleton is called the underlying graph of Δ. The 0-skeleton of Δ can be identified with its vertexset, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the0-skeleton is a family of single-element sets).The link of a face Y in Δ, often denoted Δ/Y or lkΔ(Y), is the subcomplex of Δ defined by

∆/Y := X ∈ ∆ | X ∩ Y = ∅, X ∪ Y ∈ ∆.

Note that the link of the empty set is Δ itself.

1

2 CHAPTER 1. ABSTRACT SIMPLICIAL COMPLEX

Given two abstract simplicial complexes, Δ and Γ, a simplicial map is a function f that maps the vertices of Δ to thevertices of Γ and that has the property that for any face X of Δ, the image set f (X) is a face of Γ.

1.2 Geometric realization

We can associate to an abstract simplicial complex K a topological space |K |, called its geometric realization, whichis a simplicial complex. The construction goes as follows.First, define |K | as a subset of [0, 1]S consisting of functions t : S → [0, 1] satisfying the two conditions:

∑s∈S

ts = 1

s ∈ S : ts > 0 ∈ ∆

Now think of [0, 1]S as the direct limit of [0, 1]A where A ranges over finite subsets of S, and give [0, 1]S the inducedtopology. Now give |K | the subspace topology.Alternatively, let K denote the category whose objects are the faces of K and whose morphisms are inclusions. Nextchoose a total order on the vertex set of K and define a functor F from K to the category of topological spaces asfollows. For any face X ∈ K of dimension n, let F(X) = Δn be the standard n-simplex. The order on the vertex setthen specifies a unique bijection between the elements of X and vertices of Δn, ordered in the usual way e0 < e1 < ...< en. If Y ⊂ X is a face of dimension m < n, then this bijection specifies a unique m-dimensional face of Δn. DefineF(Y) → F(X) to be the unique affine linear embedding of Δm as that distinguished face of Δn, such that the map onvertices is order preserving.We can then define the geometric realization |K | as the colimit of the functor F. More specifically |K | is the quotientspace of the disjoint union

⨿X∈K

F (X)

by the equivalence relation which identifies a point y ∈ F(Y) with its image under the map F(Y) → F(X), for everyinclusion Y ⊂ X.If K is finite, then we can describe |K | more simply. Choose an embedding of the vertex set of K as an affinelyindependent subset of some Euclidean space RN of sufficiently high dimension N. Then any face X ∈ K can beidentified with the geometric simplex in RN spanned by the corresponding embedded vertices. Take |K | to be theunion of all such simplices.If K is the standard combinatorial n-simplex, then |K | can be naturally identified with Δn.

1.3 Examples• As an example, let V be a finite subset of S of cardinality n + 1 and let K be the power set of V. Then K is called

a combinatorial n-simplex with vertex set V. If V = S = 0, 1, ..., n, K is called the standard combinatorialn-simplex.

• The clique complex of an undirected graph has a simplex for each clique (complete subgraph) of the givengraph. Clique complexes form the prototypical example of flag complexes, complexes with the property thatevery set of elements that pairwise belong to simplexes of the complex is itself a simplex.

• In the theory of partially ordered sets (“posets”), the order complex of a poset is the set of all finite chains.Its homology groups and other topological invariants contain important information about the poset.

• The Vietoris–Rips complex is defined from any metric space M and distance δ by forming a simplex for everyfinite subset of M with diameter at most δ. It has applications in homology theory, hyperbolic groups, imageprocessing, and mobile ad hoc networking. It is another example of a flag complex.

1.4. ENUMERATION 3

1.4 Enumeration

The number of abstract simplicial complexes on up to n elements is one less than the nth Dedekind number. Thesenumbers grow very rapidly, and are known only for n ≤ 8; they are (starting with n = 0):

1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787 (sequence A014466 inOEIS). This corresponds to the number of nonempty antichains of subsets of an n set.

The number of abstract simplicial complexes on exactly n labeled elements is given by the sequence “1, 2, 9, 114,6894, 7785062, 2414627396434, 56130437209370320359966” (sequence A006126 in OEIS), starting at n = 1.This corresponds to the number of antichain covers of a labeled n-set; there is a clear bijection between antichaincovers of an n-set and simplicial complexes on n elements described in terms of their maximal faces.The number of abstract simplicial complexes on exactly n unlabeled elements is given by the sequence “1, 2, 5, 20,180, 16143” (sequence A006602 in OEIS) , starting at n = 1.

1.5 See also• Kruskal–Katona theorem

1.6 References[1] Lee, JM, Introduction to Topological Manifolds, Springer 2011, ISBN 1-4419-7939-5, p153

[2] Korte, Bernhard; Lovász, László; Schrader, Rainer (1991). Greedoids. Springer-Verlag. p. 9. ISBN 3-540-18190-3.

4 CHAPTER 1. ABSTRACT SIMPLICIAL COMPLEX

A geometrical representation of an abstract simplicial complex that is not a valid simplicial complex.

Chapter 2

Almost disjoint sets

In mathematics, two sets are almost disjoint [1][2] if their intersection is small in some sense; different definitions of“small” will result in different definitions of “almost disjoint”.

2.1 Definition

The most common choice is to take “small” to mean finite. In this case, two sets are almost disjoint if their intersectionis finite, i.e. if

|A ∩B| <∞.

(Here, '|X|' denotes the cardinality of X, and '< ∞' means 'finite'.) For example, the closed intervals [0, 1] and [1,2] are almost disjoint, because their intersection is the finite set 1. However, the unit interval [0, 1] and the set ofrational numbers Q are not almost disjoint, because their intersection is infinite.This definition extends to any collection of sets. A collection of sets is pairwise almost disjoint or mutually almostdisjoint if any two distinct sets in the collection are almost disjoint. Often the prefix “pairwise” is dropped, and apairwise almost disjoint collection is simply called “almost disjoint”.Formally, let I be an index set, and for each i in I, let Ai be a set. Then the collection of sets Ai : i in I is almostdisjoint if for any i and j in I,

Ai = Aj ⇒ |Ai ∩Aj | <∞.

For example, the collection of all lines through the origin in R2 is almost disjoint, because any two of them only meetat the origin. If Ai is an almost disjoint collection consisting of more than one set, then clearly its intersection isfinite:

∩i∈I

Ai <∞.

However, the converse is not true—the intersection of the collection

1, 2, 3, . . ., 2, 3, 4, . . ., 3, 4, 5, . . ., . . .

is empty, but the collection is not almost disjoint; in fact, the intersection of any two distinct sets in this collection isinfinite.

5

6 CHAPTER 2. ALMOST DISJOINT SETS

2.2 Other meanings

Sometimes “almost disjoint” is used in some other sense, or in the sense of measure theory or topological category.Here are some alternative definitions of “almost disjoint” that are sometimes used (similar definitions apply to infinitecollections):

• Let κ be any cardinal number. Then two sets A and B are almost disjoint if the cardinality of their intersectionis less than κ, i.e. if

|A ∩B| < κ.

The case of κ = 1 is simply the definition of disjoint sets; the case of

κ = ℵ0

is simply the definition of almost disjoint given above, where the intersection of A and B is finite.

• Let m be a complete measure on a measure space X. Then two subsets A and B of X are almost disjoint if theirintersection is a null-set, i.e. if

m(A ∩B) = 0.

• Let X be a topological space. Then two subsets A and B of X are almost disjoint if their intersection is meagrein X.

2.3 References[1] Kunen, K. (1980), “Set Theory; an introduction to independence proofs”, North Holland, p. 47

[2] Jech, R. (2006) “Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118

Chapter 3

Antimatroid

a,b

a,b,c

a,c b,c

a c

Ø

«abc»«ab»«»«a»

«ac»«acb»«c»«ca»«cab»«cb»«cba»

a c

a,b b,c

a,b,c,d

«abcd»

«acbd»

«cabd»

«cbad»

a,b,c,d

Three views of an antimatroid: an inclusion ordering on its family of feasible sets, a formal language, and the corresponding pathposet.

In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by includingelements one at a time, and in which an element, once available for inclusion, remains available until it is included.Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible statesof such a process, or as a formal language modeling the different sequences in which elements may be included.Dilworth (1940) was the first to study antimatroids, using yet another axiomatization based on lattice theory, andthey have been frequently rediscovered in other contexts;[1] see Korte et al. (1991) for a comprehensive survey ofantimatroid theory with many additional references.The axioms defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are definedby an exchange axiom (e.g., the basis exchange, or independent set exchange axioms), antimatroids are defined insteadby an anti-exchange axiom, from which their name derives. Antimatroids can be viewed as a special case of greedoidsand of semimodular lattices, and as a generalization of partial orders and of distributive lattices. Antimatroids areequivalent, by complementation, to convex geometries, a combinatorial abstraction of convex sets in geometry.

7

8 CHAPTER 3. ANTIMATROID

Antimatroids have been applied to model precedence constraints in scheduling problems, potential event sequencesin simulations, task planning in artificial intelligence, and the states of knowledge of human learners.

3.1 Definitions

An antimatroid can be defined as a finite family F of sets, called feasible sets, with the following two properties:

• The union of any two feasible sets is also feasible. That is, F is closed under unions.

• If S is a nonempty feasible set, then there exists some x in S such that S \ x (the set formed by removing xfrom S) is also feasible. That is, F is an accessible set system.

Antimatroids also have an equivalent definition as a formal language, that is, as a set of strings defined from a finitealphabet of symbols. A language L defining an antimatroid must satisfy the following properties:

• Every symbol of the alphabet occurs in at least one word of L.

• Each word of L contains at most one copy of any symbol.

• Every prefix of a string in L is also in L.

• If s and t are strings in L, and s contains at least one symbol that is not in t, then there is a symbol x in s suchthat tx is another string in L.

If L is an antimatroid defined as a formal language, then the sets of symbols in strings of L form an accessible union-closed set system. In the other direction, if F is an accessible union-closed set system, and L is the language of stringss with the property that the set of symbols in each prefix of s is feasible, then L defines an antimatroid. Thus, thesetwo definitions lead to mathematically equivalent classes of objects.[2]

3.2 Examples

• A chain antimatroid has as its formal language the prefixes of a single word, and as its feasible sets the setsof symbols in these prefixes. For instance the chain antimatroid defined by the word “abcd” has as its formallanguage the strings ε, “a”, “ab”, “abc”, “abcd" and as its feasible sets the sets Ø, a, a,b, a,b,c, anda,b,c,d.[3]

• A poset antimatroid has as its feasible sets the lower sets of a finite partially ordered set. By Birkhoff’s rep-resentation theorem for distributive lattices, the feasible sets in a poset antimatroid (ordered by set inclusion)form a distributive lattice, and any distributive lattice can be formed in this way. Thus, antimatroids can beseen as generalizations of distributive lattices. A chain antimatroid is the special case of a poset antimatroidfor a total order.[3]

• A shelling sequence of a finite set U of points in the Euclidean plane or a higher-dimensional Euclidean spaceis an ordering on the points such that, for each point p, there is a line (in the Euclidean plane, or a hyperplanein a Euclidean space) that separates p from all later points in the sequence. Equivalently, p must be a vertexof the convex hull of it and all later points. The partial shelling sequences of a point set form an antimatroid,called a shelling antimatroid. The feasible sets of the shelling antimatroid are the intersections of U with thecomplement of a convex set.[3]

• A perfect elimination ordering of a chordal graph is an ordering of its vertices such that, for each vertex v,the neighbors of v that occur later than v in the ordering form a clique. The prefixes of perfect eliminationorderings of a chordal graph form an antimatroid.[3]

3.3. PATHS AND BASIC WORDS 9

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

A shelling sequence of a planar point set. The line segments show edges of the convex hulls after some of the points have beenremoved.

3.3 Paths and basic words

In the set theoretic axiomatization of an antimatroid there are certain special sets called paths that determine thewhole antimatroid, in the sense that the sets of the antimatroid are exactly the unions of paths. If S is any feasible setof the antimatroid, an element x that can be removed from S to form another feasible set is called an endpoint of S,and a feasible set that has only one endpoint is called a path of the antimatroid. The family of paths can be partiallyordered by set inclusion, forming the path poset of the antimatroid.For every feasible set S in the antimatroid, and every element x of S, one may find a path subset of S for which xis an endpoint: to do so, remove one at a time elements other than x until no such removal leaves a feasible subset.Therefore, each feasible set in an antimatroid is the union of its path subsets. If S is not a path, each subset in thisunion is a proper subset of S. But, if S is itself a path with endpoint x, each proper subset of S that belongs to theantimatroid excludes x. Therefore, the paths of an antimatroid are exactly the sets that do not equal the unions oftheir proper subsets in the antimatroid. Equivalently, a given family of sets P forms the set of paths of an antimatroidif and only if, for each S in P, the union of subsets of S in P has one fewer element than S itself. If so, F itself is thefamily of unions of subsets of P.In the formal language formalization of an antimatroid we may also identify a subset of words that determine the wholelanguage, the basic words. The longest strings in L are called basic words; each basic word forms a permutation ofthe whole alphabet. For instance, the basic words of a poset antimatroid are the linear extensions of the given partialorder. If B is the set of basic words, L can be defined from B as the set of prefixes of words in B. It is often convenientto define antimatroids from basic words in this way, but it is not straightforward to write an axiomatic definition of

10 CHAPTER 3. ANTIMATROID

antimatroids in terms of their basic words.

3.4 Convex geometries

See also: Convex set, Convex geometry and Closure operator

If F is the set system defining an antimatroid, with U equal to the union of the sets in F, then the family of sets

G = U \ S | S ∈ F

complementary to the sets in F is sometimes called a convex geometry and the sets in G are called convex sets. Forinstance, in a shelling antimatroid, the convex sets are intersections of U with convex subsets of the Euclidean spaceinto which U is embedded.Complementarily to the properties of set systems that define antimatroids, the set system defining a convex geometryshould be closed under intersections, and for any set S in G that is not equal to U there must be an element x not in Sthat can be added to S to form another set in G.A convex geometry can also be defined in terms of a closure operator τ that maps any subset of U to its minimalclosed superset. To be a closure operator, τ should have the following properties:

• τ(∅) = ∅: the closure of the empty set is empty.

• Any set S is a subset of τ(S).

• If S is a subset of T, then τ(S) must be a subset of τ(T).

• For any set S, τ(S) = τ(τ(S)).

The family of closed sets resulting from a closure operation of this type is necessarily closed under intersections. Theclosure operators that define convex geometries also satisfy an additional anti-exchange axiom:

• If neither y nor z belong to τ(S), but z belongs to τ(S ∪ y), then y does not belong to τ(S ∪ z).

A closure operation satisfying this axiom is called an anti-exchange closure. If S is a closed set in an anti-exchangeclosure, then the anti-exchange axiom determines a partial order on the elements not belonging to S, where x ≤ y inthe partial order when x belongs to τ(S ∪ y). If x is a minimal element of this partial order, then S ∪ x is closed.That is, the family of closed sets of an anti-exchange closure has the property that for any set other than the universalset there is an element x that can be added to it to produce another closed set. This property is complementary tothe accessibility property of antimatroids, and the fact that intersections of closed sets are closed is complementaryto the property that unions of feasible sets in an antimatroid are feasible. Therefore, the complements of the closedsets of any anti-exchange closure form an antimatroid.[4]

3.5 Join-distributive lattices

Any two sets in an antimatroid have a unique least upper bound (their union) and a unique greatest lower bound(the union of the sets in the antimatroid that are contained in both of them). Therefore, the sets of an antimatroid,partially ordered by set inclusion, form a lattice. Various important features of an antimatroid can be interpreted inlattice-theoretic terms; for instance the paths of an antimatroid are the join-irreducible elements of the correspondinglattice, and the basic words of the antimatroid correspond to maximal chains in the lattice. The lattices that arise fromantimatroids in this way generalize the finite distributive lattices, and can be characterized in several different ways.

• The description originally considered by Dilworth (1940) concerns meet-irreducible elements of the lattice.For each element x of an antimatroid, there exists a unique maximal feasible set Sx that does not contain x (Sxis the union of all feasible sets not containing x). Sx is meet-irreducible, meaning that it is not the meet of any

3.6. SUPERSOLVABLE ANTIMATROIDS 11

two larger lattice elements: any larger feasible set, and any intersection of larger feasible sets, contains x and sodoes not equal Sx. Any element of any lattice can be decomposed as a meet of meet-irreducible sets, often inmultiple ways, but in the lattice corresponding to an antimatroid each element T has a unique minimal familyof meet-irreducible sets Sx whose meet is T ; this family consists of the sets Sx such that T ∪ x belongs to theantimatroid. That is, the lattice has unique meet-irreducible decompositions.

• A second characterization concerns the intervals in the lattice, the sublattices defined by a pair of lattice elementsx ≤ y and consisting of all lattice elements z with x ≤ z ≤ y. An interval is atomistic if every element in it is thejoin of atoms (the minimal elements above the bottom element x), and it is Boolean if it is isomorphic to thelattice of all subsets of a finite set. For an antimatroid, every interval that is atomistic is also boolean.

• Thirdly, the lattices arising from antimatroids are semimodular lattices, lattices that satisfy the upper semimod-ular law that for any two elements x and y, if y covers x ∧ y then x ∨ y covers x. Translating this conditioninto the sets of an antimatroid, if a set Y has only one element not belonging to X then that one element maybe added to X to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: for all lattice elements x, y, and z, if x ∧ y and x ∧ z are both equal then they alsoequal x ∧ (y ∨ z). A semimodular and meet-semidistributive lattice is called a join-distributive lattice.

These three characterizations are equivalent: any lattice with unique meet-irreducible decompositions has booleanatomistic intervals and is join-distributive, any lattice with boolean atomistic intervals has unique meet-irreducibledecompositions and is join-distributive, and any join-distributive lattice has unique meet-irreducible decompositionsand boolean atomistic intervals.[5] Thus, we may refer to a lattice with any of these three properties as join-distributive.Any antimatroid gives rise to a finite join-distributive lattice, and any finite join-distributive lattice comes from anantimatroid in this way.[6] Another equivalent characterization of finite join-distributive lattices is that they are graded(any two maximal chains have the same length), and the length of a maximal chain equals the number of meet-irreducible elements of the lattice.[7] The antimatroid representing a finite join-distributive lattice can be recoveredfrom the lattice: the elements of the antimatroid can be taken to be the meet-irreducible elements of the lattice, andthe feasible set corresponding to any element x of the lattice consists of the set of meet-irreducible elements y suchthat y is not greater than or equal to x in the lattice.This representation of any finite join-distributive lattice as an accessible family of sets closed under unions (that is, asan antimatroid) may be viewed as an analogue of Birkhoff’s representation theorem under which any finite distributivelattice has a representation as a family of sets closed under unions and intersections.

3.6 Supersolvable antimatroids

Motivated by a problem of defining partial orders on the elements of a Coxeter group, Armstrong (2007) studied an-timatroids which are also supersolvable lattices. A supersolvable antimatroid is defined by a totally ordered collectionof elements, and a family of sets of these elements. The family must include the empty set. Additionally, it musthave the property that if two sets A and B belong to the family, the set-theoretic difference B \ A is nonempty, and xis the smallest element of B \ A, then A ∪ x also belongs to the family. As Armstrong observes, any family of setsof this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroidsthat this construction can form.

3.7 Join operation and convex dimension

If A and B are two antimatroids, both described as a family of sets, and if the maximal sets in A and B are equal, wecan form another antimatroid, the join of A and B, as follows:

A ∨B = S ∪ T | S ∈ A ∧ T ∈ B.

This is a different operation than the join considered in the lattice-theoretic characterizations of antimatroids: itcombines two antimatroids to form another antimatroid, rather than combining two sets in an antimatroid to formanother set. The family of all antimatroids that have a given maximal set forms a semilattice with this join operation.

12 CHAPTER 3. ANTIMATROID

Joins are closely related to a closure operation that maps formal languages to antimatroids, where the closure of alanguage L is the intersection of all antimatroids containing L as a sublanguage. This closure has as its feasible setsthe unions of prefixes of strings in L. In terms of this closure operation, the join is the closure of the union of thelanguages of A and B.Every antimatroid can be represented as a join of a family of chain antimatroids, or equivalently as the closure ofa set of basic words; the convex dimension of an antimatroid A is the minimum number of chain antimatroids (orequivalently the minimum number of basic words) in such a representation. If F is a family of chain antimatroidswhose basic words all belong to A, then F generates A if and only if the feasible sets of F include all paths of A. Thepaths of A belonging to a single chain antimatroid must form a chain in the path poset of A, so the convex dimensionof an antimatroid equals the minimum number of chains needed to cover the path poset, which by Dilworth’s theoremequals the width of the path poset.[8]

If one has a representation of an antimatroid as the closure of a set of d basic words, then this representation canbe used to map the feasible sets of the antimatroid into d-dimensional Euclidean space: assign one coordinate perbasic word w, and make the coordinate value of a feasible set S be the length of the longest prefix of w that is asubset of S. With this embedding, S is a subset of T if and only if the coordinates for S are all less than or equal tothe corresponding coordinates of T. Therefore, the order dimension of the inclusion ordering of the feasible sets isat most equal to the convex dimension of the antimatroid.[9] However, in general these two dimensions may be verydifferent: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.

3.8 Enumeration

The number of possible antimatroids on a set of elements grows rapidly with the number of elements in the set. Forsets of one, two, three, etc. elements, the number of distinct antimatroids is

1, 3, 22, 485, 59386, 133059751, ... (sequence A119770 in OEIS).

3.9 Applications

Both the precedence and release time constraints in the standard notation for theoretic scheduling problems maybe modeled by antimatroids. Boyd & Faigle (1990) use antimatroids to generalize a greedy algorithm of EugeneLawler for optimally solving single-processor scheduling problems with precedence constraints in which the goal isto minimize the maximum penalty incurred by the late scheduling of a task.Glasserman & Yao (1994) use antimatroids to model the ordering of events in discrete event simulation systems.Parmar (2003) uses antimatroids to model progress towards a goal in artificial intelligence planning problems.In mathematical psychology, antimatroids have been used to describe feasible states of knowledge of a human learner.Each element of the antimatroid represents a concept that is to be understood by the learner, or a class of problems thathe or she might be able to solve correctly, and the sets of elements that form the antimatroid represent possible sets ofconcepts that could be understood by a single person. The axioms defining an antimatroid may be phrased informallyas stating that learning one concept can never prevent the learner from learning another concept, and that any feasiblestate of knowledge can be reached by learning a single concept at a time. The task of a knowledge assessment systemis to infer the set of concepts known by a given learner by analyzing his or her responses to a small and well-chosenset of problems. In this context antimatroids have also been called “learning spaces” and “well-graded knowledgespaces”.[10]

3.10 Notes

[1] Two early references are Edelman (1980) and Jamison (1980); Jamison was the first to use the term “antimatroid”.Monjardet (1985) surveys the history of rediscovery of antimatroids.

[2] Korte et al., Theorem 1.4.

3.11. REFERENCES 13

[3] Gordon (1997) describes several results related to antimatroids of this type, but these antimatroids were mentioned earliere.g. by Korte et al. Chandran et al. (2003) use the connection to antimatroids as part of an algorithm for efficiently listingall perfect elimination orderings of a given chordal graph.

[4] Korte et al., Theorem 1.1.

[5] Adaricheva, Gorbunov & Tumanov (2003), Theorems 1.7 and 1.9; Armstrong (2007), Theorem 2.7.

[6] Edelman (1980), Theorem 3.3; Armstrong (2007), Theorem 2.8.

[7] Monjardet (1985) credits a dual form of this characterization to several papers from the 1960s by S. P. Avann.

[8] Edelman & Saks (1988); Korte et al., Theorem 6.9.

[9] Korte et al., Corollary 6.10.

[10] Doignon & Falmagne (1999).

3.11 References• Adaricheva, K. V.; Gorbunov, V. A.; Tumanov, V. I. (2003), “Join-semidistributive lattices and convex ge-

ometries”, Advances in Mathematics 173 (1): 1–49, doi:10.1016/S0001-8708(02)00011-7.

• Armstrong, Drew (2007), The sorting order on a Coxeter group, arXiv:0712.1047.

• Birkhoff, Garrett; Bennett, M. K. (1985), “The convexity lattice of a poset”,Order 2 (3): 223–242, doi:10.1007/BF00333128.

• Björner, Anders; Ziegler, Günter M. (1992), “Introduction to greedoids”, in White, Neil,Matroid Applications,Encyclopedia of Mathematics and its Applications 40, Cambridge: Cambridge University Press, pp. 284–357,doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537

• Boyd, E. Andrew; Faigle, Ulrich (1990), “An algorithmic characterization of antimatroids”, Discrete AppliedMathematics 28 (3): 197–205, doi:10.1016/0166-218X(90)90002-T.

• Chandran, L. S.; Ibarra, L.; Ruskey, F.; Sawada, J. (2003), “Generating and characterizing the perfect elimina-tion orderings of a chordal graph” (PDF), Theoretical Computer Science 307 (2): 303–317, doi:10.1016/S0304-3975(03)00221-4.

• Dilworth, Robert P. (1940), “Lattices with unique irreducible decompositions”, Annals of Mathematics 41 (4):771–777, doi:10.2307/1968857, JSTOR 1968857.

• Doignon, Jean-Paul; Falmagne, Jean-Claude (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2.

• Edelman, Paul H. (1980), “Meet-distributive lattices and the anti-exchange closure”, Algebra Universalis 10(1): 290–299, doi:10.1007/BF02482912.

• Edelman, Paul H.; Saks, Michael E. (1988), “Combinatorial representation and convex dimension of convexgeometries”, Order 5 (1): 23–32, doi:10.1007/BF00143895.

• Glasserman, Paul; Yao, David D. (1994), Monotone Structure in Discrete Event Systems, Wiley Series in Prob-ability and Statistics, Wiley Interscience, ISBN 978-0-471-58041-6.

• Gordon, Gary (1997), “A β invariant for greedoids and antimatroids”, Electronic Journal of Combinatorics 4(1): Research Paper 13, MR 1445628.

• Jamison, Robert (1980), “Copoints in antimatroids”, Proceedings of the Eleventh Southeastern Conference onCombinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1980), Vol. II, Con-gressus Numerantium 29, pp. 535–544, MR 608454.

• Korte, Bernhard; Lovász, László; Schrader, Rainer (1991), Greedoids, Springer-Verlag, pp. 19–43, ISBN3-540-18190-3.

• Monjardet, Bernard (1985), “A use for frequently rediscovering a concept”,Order 1 (4): 415–417, doi:10.1007/BF00582748.

• Parmar, Aarati (2003), “Some Mathematical Structures Underlying Efficient Planning”, AAAI Spring Sympo-sium on Logical Formalization of Commonsense Reasoning (PDF).

Chapter 4

Block design

This article is about block designs with fixed block size (uniform). For block designs with variable block sizes, seeCombinatorial design. For experimental designs in statistics, see randomized block design.

In combinatorial mathematics, a block design is a set together with a family of subsets (repeated subsets are allowedat times) whose members are chosen to satisfy some set of properties that are deemed useful for a particular appli-cation. These applications come from many areas, including experimental design, finite geometry, software testing,cryptography, and algebraic geometry. Many variations have been examined, but the most intensely studied are thebalanced incomplete block designs (BIBDs or 2-designs) which historically were related to statistical issues in thedesign of experiments.[1][2]

A block design in which all the blocks have the same size is called uniform. The designs discussed in this article areall uniform. Pairwise balanced designs (PBDs) are examples of block designs that are not necessarily uniform.

4.1 Definition of a BIBD (or 2-design)

Given a finite set X (of elements called points) and integers k, r, λ ≥ 1, we define a 2-design (or BIBD, standing forbalanced incomplete block design) B to be a family of k-element subsets of X, called blocks, such that the numberr of blocks containing x in X is not dependent on which x is chosen, and the number λ of blocks containing givendistinct points x and y in X is also independent of the choices.“Family” in the above definition can be replaced by “set” if repeated blocks are not allowed. Designs in which repeatedblocks are not allowed are called simple.Here v (the number of elements of X, called points), b (the number of blocks), k, r, and λ are the parameters of thedesign. (To avoid degenerate examples, it is also assumed that v > k, so that no block contains all the elements of theset. This is the meaning of “incomplete” in the name of these designs.) In a table:

The design is called a (v, k, λ)-design or a (v, b, r, k, λ)-design. The parameters are not all independent; v, k, andλ determine b and r, and not all combinations of v, k, and λ are possible. The two basic equations connecting theseparameters are

bk = vr,

λ(v − 1) = r(k − 1).

These conditions are not sufficient as, for example, a (43,7,1)-design does not exist.[3]

The order of a 2-design is defined to be n = r − λ. The complement of a 2-design is obtained by replacing each blockwith its complement in the point set X. It is also a 2-design and has parameters v′ = v, b′ = b, r′ = b − r, k′ = v − k, λ′= λ + b − 2r. A 2-design and its complement have the same order.A fundamental theorem, Fisher’s inequality, named after the statistician Ronald Fisher, is that b ≥ v in any 2-design.

14

4.2. SYMMETRIC BIBDS 15

4.2 Symmetric BIBDs

The case of equality in Fisher’s inequality, that is, a 2-design with an equal number of points and blocks, is calleda symmetric design.[4] Symmetric designs have the smallest number of blocks amongst all the 2-designs with thesame number of points.In a symmetric design r = k holds as well as b = v, and, while it is generally not true in arbitrary 2-designs, in asymmetric design every two distinct blocks meet in λ points.[5] A theorem of Ryser provides the converse. If X isa v-element set, and B is a v-element set of k-element subsets (the “blocks”), such that any two distinct blocks haveexactly λ points in common, then (X, B) is a symmetric block design.[6]

The parameters of a symmetric design satisfy

λ(v − 1) = k(k − 1).

This imposes strong restrictions on v, so the number of points is far from arbitrary. The Bruck–Ryser–Chowlatheorem gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of theseparameters.The following are important examples of symmetric 2-designs:

4.2.1 Projective planes

Main article: Projective plane

Finite projective planes are symmetric 2-designs with λ = 1 and order n > 1. For these designs the symmetric designequation becomes:

v − 1 = k(k − 1).

Since k = r we can write the order of a projective plane as n = k − 1 and, from the displayed equation above, we obtainv = (n + 1)n + 1 = n2 + n + 1 points in a projective plane of order n.As a projective plane is a symmetric design, we have b = v, meaning that b = n2 + n + 1 also. The number b is thenumber of lines of the projective plane. There can be no repeated lines since λ = 1, so a projective plane is a simple2-design in which the number of lines and the number of points are always the same. For a projective plane, k is thenumber of points on each line and it is equal to n + 1. Similarly, r = n + 1 is the number of lines with which a givenpoint is incident.For n = 2 we get a projective plane of order 2, also called the Fano plane, with v = 4 + 2 + 1 = 7 points and 7 lines.In the Fano plane, each line has n + 1 = 3 points and each point belongs to n + 1 = 3 lines.Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the onlyknown infinite family (with respect to having a constant λ value) of symmetric block designs.[7]

4.2.2 Biplanes

A biplane or biplane geometry is a symmetric 2-design with λ = 2; that is, every set of two points is contained intwo blocks (“lines”), while any two lines intersect in two points.[7] They are similar to finite projective planes, exceptthat rather than two points determining one line (and two lines determining one point), two points determine twolines (respectively, points). A biplane of order n is one whose blocks have k = n + 2 points; it has v = 1 + (n + 2)(n +1)/2 points (since r = k).The 18 known examples[8] are listed below.

• (Trivial) The order 0 biplane has 2 points (and lines of size 2; a 2-(2,2,2) design); it is two points, with twoblocks, each consisting of both points. Geometrically, it is the digon.

16 CHAPTER 4. BLOCK DESIGN

• The order 1 biplane has 4 points (and lines of size 3; a 2-(4,3,2) design); it is the complete design with v = 4and k = 3. Geometrically, the points are the vertices and the blocks are the faces of a tetrahedron.

• The order 2 biplane is the complement of the Fano plane: it has 7 points (and lines of size 4; a 2-(7,4,2)),where the lines are given as the complements of the (3-point) lines in the Fano plane.[9]

• The order 3 biplane has 11 points (and lines of size 5; a 2-(11,5,2)), and is also known as the Paley biplaneafter Raymond Paley; it is associated to the Paley digraph of order 11, which is constructed using the field with11 elements, and is the Hadamard 2-design associated to the size 12 Hadamard matrix; see Paley constructionI.

Algebraically this corresponds to the exceptional embedding of the projective special linear groupPSL(2,5)in PSL(2,11) – see projective linear group: action on p points for details.[10]

• There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)). These three designs are alsoMenon designs.

• There are four biplanes of order 7 (and 37 points, lines of size 9; a 2-(37,9,2)).[11]

• There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).[12]

• Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).[13]

4.2.3 Hadamard 2-designs

An Hadamard matrix of size m is an m × m matrix H whose entries are ±1 such that HH⊤ = mI , where H⊤ is thetranspose of H and Im is the m × m identity matrix. An Hadamard matrix can be put into standardized form (that is,converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size m >2 then m must be a multiple of 4.Given an Hadamard matrix of size 4a in standardized form, remove the first row and first column and convert every−1 to a 0. The resulting 0–1 matrix M is the incidence matrix of a symmetric 2-(4a − 1, 2a − 1, a − 1) design calledan Hadamard 2-design.[14] This construction is reversible, and the incidence matrix of a symmetric 2-design withthese parameters can be used to form an Hadamard matrix of size 4a.

4.3 Resolvable 2-designs

A resolvable 2-design is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of whichforms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design.If a 2-(v,k,λ) resolvable design has c parallel classes, then b ≥ v + c − 1.[15]

Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.[16]

Archetypical resolvable 2-designs are the finite affine planes. A solution of the famous 15 schoolgirl problem is aresolution of a 2-(15,3,1) design.[17]

4.4 Generalization: t-designs

Given any positive integer t, a t-design B is a class of k-element subsets of X, called blocks, such that every point x inX appears in exactly r blocks, and every t-element subset T appears in exactly λ blocks. The numbers v (the numberof elements of X), b (the number of blocks), k, r, λ, and t are the parameters of the design. The design may be calleda t-(v,k,λ)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosenarbitrarily. The equations are

λi = λ

(v − i

t− i

)/(k − i

t− i

)for i = 0, 1, . . . , t,

4.5. STEINER SYSTEMS 17

where λi is the number of blocks that contain any i-element set of points.Theorem:[18] Any t-(v,k,λ)-design is also an s-(v,k,λ )-design for any s with 1 ≤ s ≤ t. (Note that the “lambda value”changes as above and depends on s.)A consequence of this theorem is that every t-design with t ≥ 2 is also a 2-design.There are no known examples of non-trivial t-(v,k,1)-designs with t > 5 .The term block design by itself usually means a 2-design.

4.4.1 Derived and extendable t-designs

Let D = (X, B) be a t-(v,k,λ) design and p a point of X. The derived design Dp has point set X − p and as block setall the blocks of D which contain p with p removed. It is a (t − 1)-(v − 1, k − 1, λ) design. Note that derived designswith respect to different points may not be isomorphic. A design E is called an extension of D if E has a point p suchthat E is isomorphic to D; we call D extendable if it has an extension.Theorem:[19] If a t-(v,k,λ) design has an extension, then k + 1 divides b(v + 1).The only extendable projective planes (symmetric 2-(n2 + n + 1, n + 1, 1) designs) are those of orders 2 and 4.[20]

Every Hadamard 2-design is extendable (to an Hadamard 3-design).[21]

Theorem:.[22] If D, a symmetric 2-(v,k,λ) design, is extendable, then one of the following holds:

1. D is an Hadamard 2-design,2. v = (λ + 2)(λ2 + 4λ + 2), k = λ2 + 3λ + 1,3. v = 495, k = 39, λ = 3.

Note that the projective plane of order two is an Hadamard 2-design; the projective plane of order four has parameterswhich fall in case 2; the only other known symmetric 2-designs with parameters in case 2 are the order 9 biplanes,but none of them are extendable; and there is no known symmetric 2-design with the parameters of case 3.[23]

Inversive planes

A design with the parameters of the extension of an affine plane, i.e., a 3-(n2 + 1, n + 1, 1) design, is called a finiteinversive plane, or Möbius plane, of order n.It is possible to give a geometric description of some inversive planes, indeed, of all known inversive planes. Anovoid in PG(3,q) is a set of q2 + 1 points, no three collinear. It can be shown that every plane (which is a hyperplanesince the geometric dimension is 3) of PG(3,q) meets an ovoid O in either 1 or q + 1 points. The plane sections ofsize q + 1 of O are the blocks of an inversive plane of order q. Any inversive plane arising this way is called egglike.All known inversive planes are egglike.An example of an ovoid is the elliptic quadric, the set of zeros of the quadratic form

x1x2 + f(x3, x4),

where f is an irreducible quadratic form in two variables over GF(q). [f(x,y) = x2 + xy + y2 for example].If q is an odd power of 2, another type of ovoid is known – the Suzuki–Tits ovoid.Theorem. Let q be a positive integer, at least 2. (a) If q is odd, then any ovoid is projectively equivalent to the ellipticquadric in a projective geometry PG(3,q); so q is a prime power and there is a unique egglike inversive plane of orderq. (But it is unknown if non-egglike ones exist.) (b) if q is even, then q is a power of 2 and any inversive plane oforder q is egglike (but there may be some unknown ovoids).

4.5 Steiner systems

Main article: Steiner system

18 CHAPTER 4. BLOCK DESIGN

A Steiner system (named after Jakob Steiner) is a t-design with λ = 1 and t ≥ 2.A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsetsof S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In the generalnotation for block designs, an S(t,k,n) would be a t-(n,k,1) design.This definition is relatively modern, generalizing the classical definition of Steiner systems which in addition requiredthat k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple system, while an S(3,4,n) was called a Steinerquadruple system, and so on. With the generalization of the definition, this naming system is no longer strictlyadhered to.Projective planes and affine planes are examples of Steiner systems under the current definition while only the Fanoplane (projective plane of order 2) would have been a Steiner system under the older definition.

4.6 Partially balanced designs (PBIBDs)

An n-class association scheme consists of a set X of size v together with a partition S of X × X into n + 1 binaryrelations, R0, R1, ..., R . A pair of elements in relation Rᵢ are said to be ith–associates. Each element of X has nᵢ ithassociates. Furthermore:

• R0 = (x, x) : x ∈ X and is called the Identity relation.

• Defining R∗ := (x, y)|(y, x) ∈ R , if R in S, then R* in S

• If (x, y) ∈ Rk , the number of z ∈ X such that (x, z) ∈ Ri and (z, y) ∈ Rj is a constant pkij depending on i,j, k but not on the particular choice of x and y.

An association scheme is commutative if pkij = pkji for all i, j and k. Most authors assume this property.A partially balanced incomplete block design with n associate classes (PBIBD(n)) is a block design based on av-set X with b blocks each of size k and with each element appearing in r blocks, such that there is an associationscheme with n classes defined on X where, if elements x and y are ith associates, 1 ≤ i ≤ n, then they are together inprecisely λᵢ blocks.A PBIBD(n) determines an association scheme but the converse is false.[24]

4.6.1 Example

Let A(3) be the following association scheme with three associate classes on the set X = 1,2,3,4,5,6. The (i,j) entryis s if elements i and j are in relation R .

The blocks of a PBIBD(3) based on A(3) are:

The parameters of this PBIBD(3) are: v = 6, b = 8, k = 3, r = 4 and λ1 = λ2 = 2 and λ3 = 1. Also, for the associationscheme we have n0 = n2 = 1 and n1 = n3 = 2.[25]

4.6.2 Properties

The parameters of a PBIBD(m) satisfy:[26]

1. vr = bk

2.∑m

i=1 ni = v − 1

4.7. APPLICATIONS 19

3.∑m

i=1 niλi = r(k − 1)

4.∑m

u=0 phju = nj

5. nipijh = njp

jih

A PBIBD(1) is a BIBD and a PBIBD(2) in which λ1 = λ2 is a BIBD.[27]

4.6.3 Two associate class PBIBDs

PBIBD(2)s have been the studied the most since they are the simplest and most useful of the PBIBDs.[28] They fallinto six types[29] based on a classification of the then known PBIBD(2)s by Bose & Shimamoto (1952):[30]

1. group divisible;

2. triangular;

3. Latin square type;

4. cyclic;

5. partial geometry type;

6. miscellaneous.

4.7 Applications

The mathematical subject of block designs originated in the statistical framework of design of experiments. Thesedesigns were especially useful in applications of the technique of analysis of variance (ANOVA). This remains asignificant area for the use of block designs.While the origins of the subject are grounded in biological applications (as is some of the existing terminology), thedesigns are used in many applications where systematic comparisons are being made, such as in software testing.The incidence matrix of block designs provide a natural source of interesting block codes that are used as errorcorrecting codes. The rows of their incidence matrices are also used as the symbols in a form of pulse-positionmodulation.[31]

4.8 See also

• Incidence geometry

4.9 Notes[1] Colbourn & Dinitz 2007

[2] Stinson 2003, pg.1

[3] Proved by Tarry in 1900 who showed that there was no pair of orthogonal Latin squares of order six. The 2-design withthe indicate parameters is equivalent to the existence of five mutually orthogonal Latin squares of order six.

[4] They have also been referred to as projective designs or square designs. These alternatives have been used in an attempt toreplace the term “symmetric”, since there is nothing symmetric (in the usual meaning of the term) about these designs. Theuse of projective is due to P.Dembowski (Finite Geometries, Springer, 1968), in analogy with the most common example,projective planes, while square is due to P. Cameron (Designs, Graphs, Codes and their Links, Cambridge, 1991) andcaptures the implication of v = b on the incidence matrix. Neither term has caught on as a replacement and these designsare still universally referred to as symmetric.

20 CHAPTER 4. BLOCK DESIGN

[5] Stinson 2003, pg.23, Theorem 2.2

[6] Ryser 1963, pp. 102–104

[7] Hughes & Piper 1985, pg.109

[8] Hall 1986, pp.320-335

[9] Assmus & Key 1992, pg.55

[10] Martin, Pablo; Singerman, David (April 17, 2008), From Biplanes to the Klein quartic and the Buckyball (PDF), p. 4

[11] Salwach & Mezzaroba 1978

[12] Kaski & Östergård 2008

[13] Aschbacher 1971, pp. 279–281

[14] Stinson 2003, pg. 74, Theorem 4.5

[15] Hughes & Piper 1985, pg. 156, Theorem 5.4

[16] Hughes & Piper 1985, pg. 158, Corollary 5.5

[17] Beth, Jungnickel & Lenz 1986, pg. 40 Example 5.8

[18] Stinson 2003, pg.203, Corollary 9.6

[19] Hughes & Piper 1985, pg.29

[20] Cameron & van Lint 1991, pg. 11, Proposition 1.34

[21] Hughes & Piper 1985, pg. 132, Theorem 4.5

[22] Cameron & van Lint 1991, pg. 11, Theorem 1.35

[23] Colbourn & Dinitz 2007, pg. 114, Remarks 6.35

[24] Street & Street 1987, pg. 237

[25] Street & Street 1987, pg. 238

[26] Street & Street 1987, pg. 240, Lemma 4

[27] Colburn & Dinitz 2007, pg. 562, Remark 42.3 (4)

[28] Street & Street 1987, pg. 242

[29] Not a mathematical classification since one of the types is a catch-all “and everything else”.

[30] Raghavarao 1988, pg. 127

[31] Noshad, Mohammad; Brandt-Pearce, Maite (Jul 2012). “Expurgated PPM Using Symmetric Balanced Incomplete BlockDesigns”. IEEE Communications Letters 16 (7): 968–971. doi:10.1109/LCOMM.2012.042512.120457.

4.10 References• Aschbacher, Michael (1971). “On collineation groups of symmetric block designs”. Journal of CombinatorialTheory, Series A 11 (3): 272–281. doi:10.1016/0097-3165(71)90054-9.

• Assmus, E.F.; Key, J.D. (1992), Designs and Their Codes, Cambridge: Cambridge University Press, ISBN0-521-41361-3

• Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge UniversityPress. 2nd ed. (1999) ISBN 978-0-521-44432-3.

• R. C. Bose, “A Note on Fisher’s Inequality for Balanced Incomplete Block Designs”, Annals of MathematicalStatistics, 1949, pages 619–620.

4.11. EXTERNAL LINKS 21

• Bose, R. C.; Shimamoto, T. (1952), “Classification and analysis of partially balanced incomplete block designswith two associate classes”, Journal of the American Statistical Association 47: 151–184, doi:10.1080/01621459.1952.10501161

• Cameron, P. J.; van Lint, J. H. (1991), Designs, Graphs, Codes and their Links, Cambridge: Cambridge Uni-versity Press, ISBN 0-521-42385-6

• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, ISBN 1-58488-506-8

• R. A. Fisher, “An examination of the different possible solutions of a problem in incomplete blocks”, Annalsof Eugenics, volume 10, 1940, pages 52–75.

• Hall, Jr., Marshall (1986), Combinatorial Theory (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-09138-3

• Hughes, D.R.; Piper, E.C. (1985), Design theory, Cambridge: Cambridge University Press, ISBN 0-521-25754-9

• Kaski, Petteri and Östergård, Patric (2008). “There Are Exactly Five Biplanes with k = 11”. Journal ofCombinatorial Designs 16 (2): 117–127. doi:10.1002/jcd.20145. MR 2008m:05038.

• Lander, E. S. (1983), Symmetric Designs: An Algebraic Approach, Cambridge: Cambridge University Press

• Lindner, C.C.; Rodger, C.A. (1997), Design Theory, Boca Raton: CRC Press, ISBN 0-8493-3986-3

• Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (correctedreprint of the 1971 Wiley ed.). New York: Dover.

• Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications.World Scientific.

• Ryser, Herbert John (1963), “Chapter 8: Combinatorial Designs”, Combinatorial Mathematics (Carus Mono-graph #14), Mathematical Association of America

• Salwach, Chester J.; Mezzaroba, Joseph A. (1978). “The four biplanes with k = 9”. Journal of CombinatorialTheory, Series A 24 (2): 141–145. doi:10.1016/0097-3165(78)90002-X.

• S. S. Shrikhande, and Vasanti N. Bhat-Nayak, Non-isomorphic solutions of some balanced incomplete blockdesigns I – Journal of Combinatorial Theory, 1970

• Stinson, Douglas R. (2003), Combinatorial Designs: Constructions and Analysis, New York: Springer, ISBN0-387-95487-2

• Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P.[Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3.

• van Lint, J.H.; Wilson, R.M. (1992). A Course in Combinatorics. Cambridge: Cambridge University Press.

4.11 External links• DesignTheory.Org: Databases of combinatorial, statistical, and experimental block designs. Software and

other resources hosted by the School of Mathematical Sciences at Queen Mary College, University of London.

• Design Theory Resources: Peter Cameron's page of web based design theory resources.

• Weisstein, Eric W., “Block Designs”, MathWorld.

Chapter 5

Carathéodory’s theorem (convex hull)

(0,1)

(0,0) (1,0)

(1,1)

(1/4,1/4)

An illustration of Carathéodory’s theorem for a square in R2

See also Carathéodory’s theorem (disambiguation) for other meanings

In convex geometry Carathéodory’s theorem states that if a point x of Rd lies in the convex hull of a set P, there

22

5.1. PROOF 23

is a subset P′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of P′. Equivalently, x liesin an r-simplex with vertices in P, where r ≤ d . The result is named for Constantin Carathéodory, who proved thetheorem in 1911 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathéodory’s theorem for anysets P in Rd.For example, consider a set P = (0,0), (0,1), (1,0), (1,1) which is a subset of R2. The convex hull of this setis a square. Consider now a point x = (1/4, 1/4), which is in the convex hull of P. We can then construct a set(0,0),(0,1),(1,0) = P′, the convex hull of which is a triangle and encloses x, and thus the theorem works for thisinstance, since |P′| = 3. It may help to visualise Carathéodory’s theorem in 2 dimensions, as saying that we canconstruct a triangle consisting of points from P that encloses any point in P.

5.1 Proof

Let x be a point in the convex hull of P. Then, x is a convex combination of a finite number of points in P :

x =

k∑j=1

λjxj

where every x is in P, every λ is non-negative, and∑k

j=1 λj = 1 .Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the points x2 − x1, ..., xk − x1 are linearly dependent,so there are real scalars μ2, ..., μk, not all zero, such that

k∑j=2

µj(xj − x1) = 0.

If μ1 is defined as

µ1 := −k∑

j=2

µj

then

k∑j=1

µjxj = 0

k∑j=1

µj = 0

and not all of the μj are equal to zero. Therefore, at least one μ > 0. Then,

x =

k∑j=1

λjxj − α

k∑j=1

µjxj =k∑

j=1

(λj − αµj)xj

for any real α. In particular, the equality will hold if α is defined as

α := min1≤j≤k

λj

µj: µj > 0

= λi

µi.

Note that α>0, and for every j between 1 and k,

24 CHAPTER 5. CARATHÉODORY’S THEOREM (CONVEX HULL)

λj − αµj ≥ 0.

In particular, λᵢ − αμi = 0 by definition of α. Therefore,

x =

k∑j=1

(λj − αµj)xj

where every λj − αµj is nonnegative, their sum is one , and furthermore, λi − αµi = 0 . In other words, x isrepresented as a convex combination of at most k−1 points of P. This process can be repeated until x is representedas a convex combination of at most d + 1 points in P.An alternative proof uses Helly’s theorem.

5.2 See also• Shapley–Folkman lemma

• Helly’s theorem

• Krein–Milman theorem

• Choquet theory

5.3 References• Carathéodory, C. (1911), "Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonis-

chen Funktionen”, Rendiconti del Circolo Matematico di Palermo 32: 193–217, doi:10.1007/BF03014795.

• Danzer, L.; Grünbaum, B.; Klee, V. (1963), “Helly’s theorem and its relatives”, Convexity, Proc. Symp. PureMath. 7, American Mathematical Society, pp. 101–179.

• Eckhoff, J. (1993), “Helly, Radon, and Carathéodory type theorems”, Handbook of Convex Geometry A, B,Amsterdam: North-Holland, pp. 389–448.

• Steinitz, Ernst (1913), “Bedingt konvergente Reihen und konvexe Systeme”, J. Reine Angew. Math. 143 (143):128–175, doi:10.1515/crll.1913.143.128.

5.4 External links• Concise statement of theorem in terms of convex hulls (at PlanetMath)

Chapter 6

Clique complex

“Whitney complex” redirects here. For the Mississippi sports facility, see Davey Whitney Complex.

The clique complex of a graph. Cliques of size one are shown as small red disks; cliques of size two are shown as black line segments;cliques of size three are shown as light blue triangles; and cliques of size four are shown as dark blue tetrahedra.

Clique complexes, flag complexes, and conformal hypergraphs are closely related mathematical objects in graphtheory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph.The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite setsclosed under the operation of taking subsets), formed by the sets of vertices in the cliques ofG. Any subset of a cliqueis itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of aset in the family should also be in the family. The clique complex can also be viewed as a topological space in whicheach clique of k vertices is represented by a simplex of dimension k − 1. The 1-skeleton of X(G) (also known as the

25

26 CHAPTER 6. CLIQUE COMPLEX

underlying graph of the complex) is an undirected graph with a vertex for every 1-element set in the family and anedge for every 2-element set in the family; it is isomorphic to G.[1]

Clique complexes are also known as Whitney complexes. A Whitney triangulation or clean triangulation of a two-dimensional manifold is an embedding of a graph G onto the manifold in such a way that every face is a triangle andevery triangle is a face. If a graph G has a Whitney triangulation, it must form a cell complex that is isomorphic to theWhitney complex of G. In this case, the complex (viewed as a topological space) is homeomorphic to the underlyingmanifold. A graph G has a 2-manifold clique complex, and can be embedded as a Whitney triangulation, if and onlyif G is locally cyclic; this means that, for every vertex v in the graph, the induced subgraph formed by the neighborsof v forms a single cycle.[2]

6.1 Independence complex

The independence complex I(G) of a graph G is formed in the same way as the clique complex from the independentsets of G. It is the clique complex of the complement graph of G.

6.2 Flag complex

In an abstract simplicial complex, a set S of vertices that is not itself part of the complex, but such that each pairof vertices in S belongs to some simplex in the complex, is called an empty simplex. Mikhail Gromov defined theno-Δ condition to be the condition that a complex have no empty simplices. A flag complex is an abstract simplicialcomplex that has no empty simplices; that is, it is a complex satisfying Gromov’s no-Δ condition. Any flag complexis the clique complex of its 1-skeleton. Thus, flag complexes and clique complexes are essentially the same thing.However, in many cases it may be convenient to define a flag complex directly from some data other than a graph,rather than indirectly as the clique complex of a graph derived from that data.[3]

6.3 Conformal hypergraph

The primal graph G(H) of a hypergraph is the graph on the same vertex set that has as its edges the pairs of verticesappearing together in the same hyperedge. A hypergraph is said to be conformal if every maximal clique of its primalgraph is a hyperedge, or equivalently, if every clique of its primal graph is contained in some hyperedge.[4] If thehypergraph is required to be downward-closed (so it contains all hyperedges that are contained in some hyperedge)then the hypergraph is conformal precisely when it is a flag complex. This relates the language of hypergraphs to thelanguage of simplicial complexes.

6.4 Examples and applications

The barycentric subdivision of any cell complex C is a flag complex having one vertex per cell of C. A collection ofvertices of the barycentric subdivision form a simplex if and only if the corresponding collection of cells of C forma flag (a chain in the inclusion ordering of the cells).[3] In particular, the barycentric subdivision of a cell complex ona 2-manifold gives rise to a Whitney triangulation of the manifold.The order complex of a partially ordered set consists of the chains (totally ordered subsets) of the partial order. Ifevery pair of some subset is itself ordered, then the whole subset is a chain, so the order complex satisfies the no-Δcondition. It may be interpreted as the clique complex of the comparability graph of the partial order.[3]

The matching complex of a graph consists of the sets of edges no two of which share an endpoint; again, this familyof sets satisfies the no-Δ condition. It may be viewed as the clique complex of the complement graph of the line graphof the given graph. When the matching complex is referred to without any particular graph as context, it means thematching complex of a complete graph. The matching complex of a complete bipartite graph Km,n is known as achessboard complex. It is the clique graph of the complement graph of a rook’s graph,[5] and each of its simplicesrepresents a placement of rooks on an m × n chess board such that no two of the rooks attack each other. When m =n ± 1, the chessboard complex forms a pseudo-manifold.

6.5. SEE ALSO 27

The Vietoris–Rips complex of a set of points in a metric space is a special case of a clique complex, formed from theunit disk graph of the points; however, every clique complex X(G) may be interpreted as the Vietoris–Rips complexof the shortest path metric on the underlying graph G.Hodkinson & Otto (2003) describe an application of conformal hypergraphs in the logics of relational structures.In that context, the Gaifman graph of a relational structure is the same as the underlying graph of the hypergraphrepresenting the structure, and a structure is guarded if it corresponds to a conformal hypergraph.Gromov showed that a cubical complex (that is, a family of hypercubes intersecting face-to-face) forms a CAT(0)space if and only if the complex is simply connected and the link of every vertex forms a flag complex. A cubicalcomplex meeting these conditions is sometimes called a cubing or a space with walls.[1][6]

6.5 See also• Simplex graph, a graph having one node for every clique of the underlying graph

• Partition matroid, a class of matroids whose intersections form clique complexes

6.6 Notes[1] Bandelt & Chepoi (2008).

[2] Hartsfeld & Ringel (1981); Larrión, Neumann-Lara & Pizaña (2002); Malnič & Mohar (1992).

[3] Davis (2002).

[4] Berge (1989); Hodkinson & Otto (2003).

[5] Dong & Wachs (2002).

[6] Chatterji & Niblo (2005).

6.7 References• Bandelt, H.-J.; Chepoi, V. (2008), “Metric graph theory and geometry: a survey”, in Goodman, J. E.; Pach,

J.; Pollack, R., Surveys on Discrete and Computational Geometry: Twenty Years Later (PDF), ContemporaryMathematics 453, Providence, RI: AMS, pp. 49–86.

• Berge, C. (1989), Hypergraphs: Combinatorics of Finite Sets, North-Holland, ISBN 0-444-87489-5.

• Chatterji, I.; Niblo, G. (2005), “From wall spaces to CAT(0) cube complexes”, International Journal of Algebraand Computation 15 (5–6): 875–885, arXiv:math.GT/0309036, doi:10.1142/S0218196705002669.

• Davis, M. W. (2002), “Nonpositive curvature and reflection groups”, in Daverman, R. J.; Sher, R. B.,Handbookof Geometric Topology, Elsevier, pp. 373–422.

• Dong, X.; Wachs, M. L. (2002), “Combinatorial Laplacian of the matching complex”, Electronic Journal ofCombinatorics 9: R17.

• Hartsfeld, N.; Ringel, Gerhard (1991), “Clean triangulations”,Combinatorica 11 (2): 145–155, doi:10.1007/BF01206358.

• Hodkinson, I.; Otto, M. (2003), “Finite conformal hypergraph covers and Gaifman cliques in finite structures”,The Bulletin of Symbolic Logic 9 (3): 387–405, doi:10.2178/bsl/1058448678.

• Larrión, F.; Neumann-Lara, V.; Pizaña, M. A. (2002), “Whitney triangulations, local girth and iterated cliquegraphs”, Discrete Mathematics 258: 123–135, doi:10.1016/S0012-365X(02)00266-2.

• Malnič, A.; Mohar, B. (1992), “Generating locally cyclic triangulations of surfaces”, Journal of CombinatorialTheory, Series B 56 (2): 147–164, doi:10.1016/0095-8956(92)90015-.

Chapter 7

Combinatorial design

Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, constructionand properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry.These concepts are not made precise so that a wide range of objects can be thought of as being under the sameumbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other timesit could involve the spatial arrangement of entries in an array as in Sudoku grids.Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory ofcombinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments.Modern applications are also found in a wide gamut of areas including; Finite geometry, tournament scheduling,lotteries, mathematical biology, algorithm design and analysis, networking, group testing and cryptography.[1]

7.1 Example

Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, eachpair of people is in exactly one set together, every two sets have exactly one person in common, and no set containseveryone, all but one person, or exactly one person? The answer depends on n.This has a solution only if n has the form q2 + q + 1. It is less simple to prove that a solution exists if q is a primepower. It is conjectured that these are the only solutions. It has been further shown that if a solution exists for qcongruent to 1 or 2 mod 4, then q is a sum of two square numbers. This last result, the Bruck–Ryser theorem, isproved by a combination of constructive methods based on finite fields and an application of quadratic forms.When such a structure does exist, it is called a finite projective plane; thus showing how finite geometry and combi-natorics intersect. When q = 2, the projective plane is called the Fano plane.

7.2 History

Combinatorial designs date to antiquity, with the Lo Shu Square being an early magic square. They developed alongwith the general growth of combinatorics from the 18th century, for example with Latin squares in the 18th cen-tury and Steiner systems in the 19th century. Designs have also been popular in recreational mathematics, such asKirkman’s schoolgirl problem (1850), and in practical problems, such as the scheduling of round-robin tournaments(solution published 1880s). In the 20th century designs were applied to the design of experiments, notably Latinsquares, finite geometry, and association schemes, yielding the field of algebraic statistics.

7.3 Fundamental combinatorial designs

The classical core of the subject of combinatorial designs is built around balanced incomplete block designs (BIBDs),Hadamard matrices and Hadamard designs, symmetric BIBDs, Latin squares, resolvable BIBDs, difference sets, andpairwise balanced designs (PBDs).[2] Other combinatorial designs are related to or have been developed from the

28

7.3. FUNDAMENTAL COMBINATORIAL DESIGNS 29

The Fano plane

study of these fundamental ones.

• A balanced incomplete block design or BIBD (usually called for short a block design) is a collection B ofb subsets (called blocks) of a finite set X of v elements, such that any element of X is contained in the samenumber r of blocks, every block has the same number k of elements, and each pair of distinct elements appeartogether in the same number λ of blocks. BIBDs are also known as 2-designs and are often denoted as 2-(v,k,λ)designs. As an example, when λ = 1 and b = v, we have a projective plane: X is the point set of the plane andthe blocks are the lines.

• A symmetric balanced incomplete block design or SBIBD is a BIBD in which v = b (the number of pointsequals the number of blocks). They are the single most important and well studied subclass of BIBDs. Pro-jective planes, biplanes and Hadamard 2-designs are all SBIBDs. They are of particular interest since they arethe extremal examples of Fisher’s inequality (b ≥ v).

• A resolvable BIBD is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of whichforms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design. Asolution of the famous 15 schoolgirl problem is a resolution of a BIBD with v = 15, k = 3 and λ = 1.[3]

• A Latin rectangle is an r × n matrix that has the numbers 1, 2, 3, ..., n as its entries (or any other set of n

30 CHAPTER 7. COMBINATORIAL DESIGN

distinct symbols) with no number occurring more than once in any row or column where r ≤ n. An n × n Latinrectangle is called a Latin square. If r < n, then it is possible to append n − r rows to an r × n Latin rectangleto form a Latin square, using Hall’s marriage theorem.[4]

Two Latin squares of order n are said to be orthogonal if the set of all ordered pairs consisting of thecorresponding entries in the two squares has n2 distinct members (all possible ordered pairs occur). Aset of Latin squares of the same order forms a set of mutually orthogonal Latin squares (MOLS)if every pair of Latin squares in the set are orthogonal. There can be at most n − 1 squares in a set ofMOLS of order n. A set of n − 1 MOLS of order n can be used to construct a projective plane of ordern (and conversely).

• A (v, k, λ) difference set is a subset D of a group G such that the order of G is v, the size of D is k, and everynonidentity element of G can be expressed as a product d1d2−1 of elements of D in exactly λ ways (when G iswritten with a multiplicative operation).[5]

If D is a difference set, and g in G, then g D = gd: d in D is also a difference set, and is called atranslate of D. The set of all translates of a difference set D forms a symmetric block design. In sucha design there are v elements and v blocks. Each block of the design consists of k points, each point iscontained in k blocks. Any two blocks have exactly λ elements in common and any two points appeartogether in λ blocks. This SBIBD is called the development of D.[6]

In particular, if λ = 1, then the difference set gives rise to a projective plane. An example of a (7,3,1)difference set in the group Z/7Z (an abelian group written additively) is the subset 1,2,4. The devel-opment of this difference set gives the Fano plane.Since every difference set gives an SBIBD, the parameter set must satisfy the Bruck–Ryser–Chowlatheorem, but not every SBIBD gives a difference set.

• AnHadamardmatrix of orderm is anm ×mmatrixHwhose entries are ±1 such thatHH⊤ = mI , whereH⊤

is the transpose of H and I is the m × m identity matrix. An Hadamard matrix can be put into standardizedform (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all+1. If the order m > 2 then m must be a multiple of 4.

Given an Hadamard matrix of order 4a in standardized form, remove the first row and first column andconvert every −1 to a 0. The resulting 0–1 matrix M is the incidence matrix of a symmetric 2 − (4a − 1,2a − 1, a − 1) design called an Hadamard 2-design.[7] This construction is reversible, and the incidencematrix of a symmetric 2-design with these parameters can be used to form an Hadamard matrix of order4a. When a = 2 we obtain the, by now familiar, Fano plane as an Hadamard 2-design.

• A pairwise balanced design (or PBD) is a set X together with a family of subsets of X (which need not havethe same size and may contain repeats) such that every pair of distinct elements of X is contained in exactly λ(a positive integer) subsets. The set X is allowed to be one of the subsets, and if all the subsets are copies of X,the PBD is called trivial. The size of X is v and the number of subsets in the family (counted with multiplicity)is b.

Fisher’s inequality holds for PBDs:[8] For any non-trivial PBD, v ≤ b.

This result also generalizes the famous Erdős–De Bruijn theorem: For a PBD with λ = 1 having no blocksof size 1 or size v, v ≤ b, with equality if and only if the PBD is a projective plane or a near-pencil.[9]

7.4 A wide assortment of other combinatorial designs

The Handbook of Combinatorial Designs (Colbourn & Dinitz 2007) has, amongst others, 65 chapters, each devotedto a combinatorial design other than those given above. A partial listing is given below:

• Association schemes

7.4. A WIDE ASSORTMENT OF OTHER COMBINATORIAL DESIGNS 31

• A balanced ternary design BTD(V, B; ρ1, ρ2, R; K, Λ) is an arrangement of V elements into B multisets(blocks), each of cardinality K (K ≤ V), satisfying:

1. Each element appears R = ρ1 + 2ρ2 times altogether, with multiplicity one in exactly ρ1 blocks and multiplicitytwo in exactly ρ2 blocks.

2. Every pair of distinct elements appears Λ times (counted with multiplicity); that is, if mᵥ is the multiplicityof the element v in block b, then for every pair of distinct elements v and w,

∑Bb=1 mvbmwb = Λ .

For example, one of the only two nonisomorphic BTD(4,8;2,3,8;4,6)s (blocks are columns) is:[10]

The incidence matrix of a BTD (where the entries are the multiplicities of the elements in the blocks)can be used to form a ternary error-correcting code analogous to the way binary codes are formed fromthe incidence matrices of BIBDs.[11]

• A balanced tournament design of order n (a BTD(n)) is an arrangement of all the distinct unordered pairsof a 2n-set V into an n × (2n − 1) array such that

1. every element of V appears precisely once in each column, and

2. every element of V appears at most twice in each row.

An example of a BTD(3) is given by

The columns of a BTD(n) provide a 1-factorization of the complete graph on 2n vertices, K₂ .[12]

BTD(n)s can be used to schedule round-robin tournaments: the rows represent the locations, the columnsthe rounds of play and the entries are the competing players or teams.

• Bent functions

• Costas arrays

• Factorial designs

• A frequency square (F-square) is a higher order generalization of a Latin square. Let S = s1,s2, ..., s be aset of distinct symbols and (λ1, λ2, ...,λ ) a frequency vector of positive integers. A frequency square of ordern is an n × n array in which each symbol sᵢ occurs λᵢ times, i = 1,2,...,m, in each row and column. The ordern = λ1 + λ2 + ... + λ . An F-square is in standard form if in the first row and column, all occurrences of sᵢprecede those of s whenever i < j.

A frequency square F1 of order n based on the set s1,s2, ..., s with frequency vector (λ1, λ2, ...,λ )and a frequency square F2, also of order n, based on the set t1,t2, ..., t with frequency vector (μ1,μ2, ...,μ ) are orthogonal if every ordered pair (sᵢ, t ) appears precisely λᵢμ times when F1 and F2 aresuperimposed.

• Hall triple systems (HTSs) are Steiner triple systems (STSs) (but the blocks are called lines) with the propertythat the substructure generated by two intersecting lines is isomorphic to the finite affine plane AG(2,3).

Any affine space AG(n,3) gives an example of an HTS. Such an HTS is an affine HTS. Nonaffine HTSsalso exist.The number of points of an HTS is 3m for some integer m ≥ 2. Nonaffine HTSs exist for any m ≥ 4 anddo not exist for m = 2 or 3.[13]

Every Steiner triple system is equivalent to a Steiner quasigroup (idempotent, commutative and satisfying(xy)y = x for all x and y). A Hall triple system is equivalent to a Steiner quasigroup which is distributive,that is, satisfies a(xy) = (ax)(ay) for all a,x,y in the quasigroup.[14]

32 CHAPTER 7. COMBINATORIAL DESIGN

• Let S be a set of 2n elements. A Howell design, H(s,2n) (on symbol set S) is an s × s array such that:

1. Each cell of the array is either empty or contains an unordered pair from S,

2. Each symbol occurs exactly once in each row and column of the array, and

3. Every unordered pair of symbols occurs in at most one cell of the array.

An example of an H(4,6) is

An H(2n − 1, 2n) is a Room square of side 2n − 1, and thus the Howell designs generalize the conceptof Room squares.

The pairs of symbols in the cells of a Howell design can be thought of as the edges of an s regular graphon 2n vertices, called the underlying graph of the Howell design.

Cyclic Howell designs are used as Howell movements in duplicate bridge tournaments. The rows of thedesign represent the rounds, the columns represent the boards, and the diagonals represent the tables.[15]

• Linear spaces

• An (n,k,p,t)-lotto design is an n-set V of elements together with a set β of k-element subsets of V (blocks),so that for any p-subset P of V, there is a block B in β for which |P ∩ B | ≥ t. L(n,k,p,t) denotes the smallestnumber of blocks in any (n,k,p,t)-lotto design. The following is a (7,5,4,3)-lotto design with the smallestpossible number of blocks:[16]

1,2,3,4,7 1,2,5,6,7 3,4,5,6,7.Lotto designs model any lottery that is run in the following way: Individuals purchase tickets consistingof k numbers chosen from a set of n numbers. At a certain point the sale of tickets is stopped and aset of p numbers is randomly selected from the n numbers. These are the winning numbers. If any soldticket contains t or more of the winning numbers, a prize is given to the ticket holder. Larger prizes goto tickets with more matches. The value of L(n,k,p,t) is of interest to both gamblers and researchers, asthis is the smallest number of tickets that are needed to be purchased in order to guarantee a prize.

The Hungarian Lottery is a (90,5,5,t)-lotto design and it is known that L(90,5,5,2) = 100. Lotterieswith parameters (49,6,6,t) are also popular worldwide and it is known that L(49,6,6,2) = 19. In generalthough, these numbers are hard to calculate and remain unknown.[17]

A geometric construction of one such design is given in Transylvanian lottery.

• Magic squares

• A (v,k,λ)-Mendelsohn design, or MD(v,k,λ),is a v-set V and a collection β of ordered k-tuples of distinctelements of V (called blocks), such that each ordered pair (x,y) with x ≠ y of elements of V is cyclicallyadjacent in λ blocks. The ordered pair (x,y) of distinct elements is cyclically adjacent in a block if the elementsappear in the block as (...,x,y,...) or (y,...,x). An MD(v,3,λ) is a Mendelsohn triple system, MTS(v,λ). Anexample of an MTS(4,1) on V = 0,1,2,3 is:

(0,1,2) (1,0,3) (2,1,3) (0,2,3)Any triple system can be made into a Mendelson triple system by replacing the unordered triple a,b,cwith the pair of ordered triples (a,b,c) and (a,c,b), but as the example shows, the converse of this state-ment is not true.If (Q,∗) is an idempotent semisymmetric quasigroup, that is, x ∗ x = x (idempotent) and x ∗ (y ∗ x) = y(semisymmetric) for all x, y in Q, let β = (x,y,x ∗ y): x, y in Q. Then (Q, β) is a Mendelsohn triplesystem MTS(|Q|,1). This construction is reversible.[18]

7.4. A WIDE ASSORTMENT OF OTHER COMBINATORIAL DESIGNS 33

• Orthogonal arrays

• A quasi-3 design is a symmetric design (SBIBD) in which each triple of blocks intersect in either x or y points,for fixed x and y called the triple intersection numbers (x < y). Any symmetric design with λ ≤ 2 is a quasi-3design with x = 0 and y = 1. The point-hyperplane design of PG(n,q) is a quasi-3 design with x = (qn−2 − 1)/(q− 1) and y = λ = (qn−1 − 1)/(q − 1). If y = λ for a quasi-3 design, the design is isomorphic to PG(n,q) or aprojective plane.[19]

• A t-(v,k,λ) design D is quasi-symmetric with intersection numbers x and y (x < y) if every two distinct blocksintersect in either x or y points. These designs naturally arise in the investigation of the duals of designs withλ = 1. A non-symmetric (b > v) 2-(v,k,1) design is quasisymmetric with x = 0 and y = 1. A multiple (repeatall blocks a certain number of times) of a symmetric 2-(v,k,λ) design is quasisymmetric with x = λ and y = k.Hadamard 3-designs (extensions of Hadamard 2-designs) are quasisymmetric.[20]

Every quasisymmetric block design gives rise to a strongly regular graph (as its block graph), but not allSRGs arise in this way.[21]

The incidence matrix of a quasisymmetric 2-(v,k,λ) design with k ≡ x ≡ y (mod 2) generates a binaryself-orthogonal code (when bordered if k is odd).[22]

• Room squares

• A spherical design is a finite set X of points in a (d − 1)-dimensional sphere such that, for some integer t, theaverage value on X of every polynomial

f(x1, . . . , xd)

of total degree at most t is equal to the average value of f on the whole sphere, i.e., the integral of fdivided by the area of the sphere.

• Turán systems

• An r × n tuscan-k rectangle on n symbols has r rows and n columns such that:

1. each row is a permutation of the n symbols and

2. for any two distinct symbols a and b and for each m from 1 to k, there is at most one row in which b is m stepsto the right of a.

If r = n and k = 1 these are referred to as Tuscan squares, while if r = n and k = n - 1 they are Florentinesquares. A Roman square is a tuscan square which is also a latin square (these are also known as rowcomplete latin squares). A Vatican square is a florentine square which is also a latin square.

The following example is a tuscan-1 square on 7 symbols which is not tuscan-2:[23]

A tuscan square on n symbols is equivalent to a decomposition of the complete graph with n verticesinto n hamiltonian directed paths.[24]

In a sequence of visual impressions, one flash card may have some effect on the impression given by thenext. This bias can be cancelled by using n sequences corresponding to the rows of an n × n tuscan-1square.[25]

• A t-wise balanced design (or t BD) of type t − (v,K,λ) is a v-set X together with a family of subsets of X(called blocks) whose sizes are in the set K, such that every t-subset of distinct elements of X is contained inexactly λ blocks. If K is a set of positive integers strictly between t and v, then the t BD is proper. If all thek-subsets of X for some k are blocks, the t BD is a trivial design.[26]

34 CHAPTER 7. COMBINATORIAL DESIGN

Notice that in the following example of a 3-12,4,6,1) design based on the set X = 1,2,...,12, somepairs appear four times (such as 1,2) while others appear five times (6,12 for instance).[27]

1 2 3 4 5 6 1 2 7 8 1 2 9 11 1 2 10 12 3 5 7 8 3 5 9 11 3 5 10 12 4 6 7 8 4 6 9 11 4 6 10 127 8 9 10 11 12 2 3 8 9 2 3 10 7 2 3 11 12 4 1 8 9 4 1 10 7 4 1 11 12 5 6 8 9 5 6 10 7 5 6 11123 4 9 10 3 4 11 8 3 4 7 12 5 2 9 10 5 2 11 8 5 2 7 12 1 6 9 10 1 6 11 8 1 6 7 124 5 10 11 4 5 7 9 4 5 8 12 1 3 10 11 1 3 7 9 1 3 8 12 2 6 10 11 2 6 7 9 2 6 8 125 1 11 7 5 1 8 10 5 1 9 12 2 4 11 7 2 4 8 10 2 4 9 12 3 6 11 7 3 6 8 10 3 6 9 12

• A Youden square is a k × v rectangular array (k < v) of v symbols such that each symbol appears exactly oncein each row and the symbols appearing in any column form a block of a symmetric (v, k, λ) design, all theblocks of which occur in this manner. A Youden square is a Latin rectangle. The term “square” in the namecomes from an older definition which did use a square array.[28] An example of a 4 × 7 Youden square is givenby:

The seven blocks (columns) form the order 2 biplane (a symmetric (7,4,2)-design).

7.5 See also• Algebraic statistics

• Hypergraph

7.6 Notes[1] Stinson 2003, pg.1

[2] Stinson 2003, pg. IX

[3] Beth, Jungnickel & Lenz 1986, pg. 40 Example 5.8

[4] Ryser 1963, pg. 52, Theorem 3.1

[5] When the group G is an abelian group (or written additively) the defining property looks like d1 –d2 from which the termdifference set comes from.

[6] Beth, Jungnickel & Lenz 1986, pg. 262, Theorem 1.6

[7] Stinson 2003, pg. 74, Theorem 4.5

[8] Stinson 2003, pg. 193, Theorem 8.20

[9] Stinson 2003, pg. 183, Theorem 8.5

[10] Colbourn & Dinitz 2007, pg. 331, Example 2.2

[11] Colbourn & Dinitz 2007, pg. 331, Remark 2.8

[12] Colbourn & Dinitz 2007, pg. 333, Remark 3.3

[13] Colbourn & Dinitz 2007, pg. 496, Theorem 28.5

[14] Colbourn & Dinitz 2007, pg. 497, Theorem 28.15

[15] Colbourn & Dinitz 2007, pg. 503, Remark 29.38

[16] Colbourn & Dinitz 2007, pg. 512, Example 32.4

[17] Colbourn & Dinitz 2007, pg. 512, Remark 32.3

7.7. REFERENCES 35

[18] Colbourn & Dinitz 2007, pg. 530, Theorem 35.15

[19] Colbourn & Dinitz 2007, pg. 577, Theorem 47.15

[20] Colbourn & Dinitz 2007, pp. 578-579

[21] Colbourn & Dinitz 2007, pg. 579, Theorem 48.10

[22] Colbourn & Dinitz 2007, pg. 580, Lemma 48.22

[23] Colbourn & Dinitz 2007, pg. 652, Examples 62.4

[24] Colbourn & Dinitz 2007, pg. 655, Theorem 62.24

[25] Colbourn & Dinitz 2007, pg. 657, Remark 62.29

[26] Colbourn & Dinitz 2007, pg. 657

[27] Colbourn & Dinitz 2007, pg. 658, Example 63.5

[28] Colbourn & Dinitz 2007, pg. 669, Remark 65.3

7.7 References• Assmus, E.F.; Key, J.D. (1992), Designs and Their Codes, Cambridge: Cambridge University Press, ISBN

0-521-41361-3

• Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge UniversityPress. 2nd ed. (1999) ISBN 978-0-521-44432-3.

• R. C. Bose, “A Note on Fisher’s Inequality for Balanced Incomplete Block Designs”, Annals of MathematicalStatistics, 1949, pages 619–620.

• Caliński, Tadeusz and Kageyama, Sanpei (2003). Block designs: A Randomization approach, Volume II: De-sign. Lecture Notes in Statistics 170. New York: Springer-Verlag. ISBN 0-387-95470-8.

• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, ISBN 1-58488-506-8

• R. A. Fisher, “An examination of the different possible solutions of a problem in incomplete blocks”, Annalsof Eugenics, volume 10, 1940, pages 52–75.

• Hall, Jr., Marshall (1986), Combinatorial Theory (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-09138-3

• Hughes, D.R.; Piper, E.C. (1985), Design theory, Cambridge: Cambridge University Press, ISBN 0-521-25754-9

• Lander, E. S. (1983), Symmetric Designs: An Algebraic Approach, Cambridge: Cambridge University Press

• Lindner, C.C.; Rodger, C.A. (1997), Design Theory, Boca Raton: CRC Press, ISBN 0-8493-3986-3

• Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (correctedreprint of the 1971 Wiley ed.). New York: Dover.

• Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications.World Scientific.

• Ryser, Herbert John (1963), “Chapter 8: Combinatorial Designs”, Combinatorial Mathematics (Carus Mono-graph #14), Mathematical Association of America

• S. S. Shrikhande, and Vasanti N. Bhat-Nayak, Non-isomorphic solutions of some balanced incomplete blockdesigns I – Journal of Combinatorial Theory, 1970

• Stinson, Douglas R. (2003), Combinatorial Designs: Constructions and Analysis, New York: Springer, ISBN0-387-95487-2

36 CHAPTER 7. COMBINATORIAL DESIGN

• Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P.[Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3.

• van Lint, J.H., and R.M. Wilson (1992), A Course in Combinatorics. Cambridge, Eng.: Cambridge UniversityPress.

7.8 External links• Design DB: A comprehensive database of combinatorial, statistical, experimental block designs

Chapter 8

Configuration (geometry)

Configurations (4362) (a complete quadrangle, at left) and (6243) (a complete quadrilateral, at right).

In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and afinite arrangement of lines, such that each point is incident to the same number of lines and each line is incident tothe same number of points.[1]

Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), theformal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his bookGeometrie der Lage, in the context of a discussion of Desargues’ theorem. Ernst Steinitz wrote his dissertation onthe subject in 1894, and they were popularized by Hilbert and Cohn-Vossen’s 1932 book Anschauliche Geometrie,reprinted in English (Hilbert & Cohn-Vossen 1952).Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclideanor projective planes (these are said to be realizable in that geometry), or as a type of abstract incidence geometry. Inthe latter case they are closely related to regular hypergraphs and biregular bipartite graphs, but with some additionalrestrictions: every two points of the incidence structure can be associated with at most one line, and every two linescan be associated with at most one point. That is, the girth of the corresponding bipartite graph (the Levi graph ofthe configuration) must be at least six.

8.1 Notation

A configuration in the plane is denoted by (pᵧ ℓπ), where p is the number of points, ℓ the number of lines, γ thenumber of lines per point, and π the number of points per line. These numbers necessarily satisfy the equation

pγ = ℓπ

37

38 CHAPTER 8. CONFIGURATION (GEOMETRY)

as this product is the number of point-line incidences (flags).Configurations having the same symbol, say (pᵧ ℓπ), need not be isomorphic as incidence structures. For instance,there exist three different (93 93) configurations: the Pappus configuration and two less notable configurations.In some configurations, p = ℓ and consequently, γ = π. These are called symmetric or balanced (Grünbaum 2009)configurations and the notation is often condensed to avoid repetition. For example (93 93) abbreviates to (93).

8.2 Examples

A (103) configuration that is not incidence-isomorphic to a Desargues configuration

Notable projective configurations include the following:

• (11), the simplest possible configuration, consisting of a point incident to a line. Often excluded as being trivial.

• (32), the triangle. Each of its three sides meets two of its three vertices, and vice versa. More generally anypolygon of n sides forms a configuration of type (n2)

• (43 62) and (62 43), the complete quadrangle and complete quadrilateral respectively.

• (73), the Fano plane. This configuration exists as an abstract incidence geometry, but cannot be constructed inthe Euclidean plane.

8.3. DUALITY OF CONFIGURATIONS 39

• (83), the Möbius–Kantor configuration. This configuration describes two quadrilaterals that are simultaneouslyinscribed and circumscribed in each other. It cannot be constructed in Euclidean plane geometry but theequations defining it have nontrivial solutions in complex numbers.

• (93), the Pappus configuration.• (94 123), the Hesse configuration of nine inflection points of a cubic curve in the complex projective plane

and the twelve lines determined by pairs of these points. This configuration shares with the Fano plane theproperty that it contains every line through its points; configurations with this property are known as Sylvester–Gallai configurations due to the Sylvester–Gallai theorem that shows that they cannot be given real-numbercoordinates (Kelly 1986).

• (103), the Desargues configuration.• (125302), the Schläfli double six, formed by 12 of the 27 lines on a cubic surface• (153), the Cremona–Richmond configuration, formed by the 15 lines complementary to a double six and their

15 tangent planes• (124 163), the Reye configuration.• (166), the Kummer configuration.• (273), the Gray configuration• (6015), the Klein configuration.

8.3 Duality of configurations

The projective dual of a configuration (pᵧ lπ) is a (lπ pᵧ) configuration in which the roles of “point” and “line” areexchanged. Types of configurations therefore come in dual pairs, except when taking the dual results in an isomorphicconfiguration. These exceptions are called self-dual configurations and in such cases p = l.[2]

8.4 The number of (n3) configurations

The number of nonisomorphic configurations of type (n3), starting at n = 7, is given by the sequence

1, 1, 3, 10, 31, 229, 2036, 21399, 245342, ... (sequence A001403 in OEIS)

These numbers count configurations as abstract incidence structures, regardless of realizability (Betten, Brinkmann& Pisanski 2000). As Gropp (1997) discusses, nine of the ten (103) configurations, and all of the (113) and (123)configurations, are realizable in the Euclidean plane, but for each n ≥ 16 there is at least one nonrealizable (n3)configuration. Gropp also points out a long-lasting error in this sequence: an 1895 paper attempted to list all (123)configurations, and found 228 of them, but the 229th configuration was not discovered until 1988.

8.5 Constructions of symmetric configurations

There are several techniques for constructing configurations, generally starting from known configurations. Some ofthe simplest of these techniques construct symmetric (pᵧ) configurations.Any finite projective plane of order n is an ((n2 + n + 1)n ₊ ₁) configuration. Let Π be a projective plane of ordern. Remove from Π a point P and all the lines of Π which pass through P (but not the points which lie on thoselines except for P) and remove a line l not passing through P and all the points that are on line l. The result is aconfiguration of type ((n2 - 1)n). If, in this construction, the line l is chosen to be a line which does pass throughP, then the construction results in a configuration of type ((n2)n). Since projective planes are known to exist for allorders n which are powers of primes, these constructions provide infinite families of symmetric configurations.Not all configurations are realizable, for instance, a (437) configuration does not exist.[3] However, Gropp (1990) hasprovided a construction which shows that for k ≥ 3, a (p ) configuration exists for all p ≥ 2 lk + 1, where lk is thelength of an optimal Golomb ruler of order k.

40 CHAPTER 8. CONFIGURATION (GEOMETRY)

8.6 Higher dimensions

The Schläfli double six.

The concept of a configuration may be generalized to higher dimensions, for instance to points and lines or planes inspace. In such cases, the restrictions that no two points belong to more than one line may be relaxed, because it ispossible for two points to belong to more than one plane.Notable three-dimensional configurations are the Möbius configuration, consisting of two mutually inscribed tetrahe-dra, Reye’s configuration, consisting of twelve points and twelve planes, with six points per plane and six planes perpoint, the Gray configuration consisting of a 3×3×3 grid of 27 points and the 27 orthogonal lines through them, andthe Schläfli double six, a configuration with 30 points, 12 lines, two lines per point, and five points per line.A further generalization is obtained in three dimensions by considering incidences of points, lines and planes, orj-spaces (0 ≤ j < 3), where each j-space is incident with Njk k-spaces (j ≠ k). Writing Njj for the number of j-spacespresent, a given configuration may be represented by the matrix:

∣∣∣∣∣∣∣∣∣∣∣∣N00 N01 N02

N10 N11 N12

N20 N21 N22

∣∣∣∣∣∣∣∣∣∣∣∣

The principle extends generally to n dimensions, where 0 ≤ j < n. Such configurations are related mathematically toregular polytopes.[4]

8.7. SEE ALSO 41

8.7 See also• Complex polytope (which could be better called complex configurations)

8.8 Notes[1] In the literature, the terms projective configuration (Hilbert & Cohn-Vossen 1952) and tactical configuration of type (1,1)

(Dembowski 1968) are also used to describe configurations as defined here.

[2] Coxeter 1999, pp. 106-149

[3] This configuration would be a projective plane of order 6 which does not exist by the Bruck-Ryser theorem.

[4] (Coxeter 1948)

8.9 References• Berman, Leah W., “Movable (n4) configurations”, The Electronic Journal of Combinatorics 13 (1): R104.

• Betten, A; Brinkmann, G.; Pisanski, T. (2000), “Counting symmetric configurations”, Discrete Applied Math-ematics 99 (1–3): 331–338, doi:10.1016/S0166-218X(99)00143-2.

• Coxeter, H.S.M. (1948), Regular Polytopes, Methuen and Co.

• Coxeter, H.S.M. (1999), “Self-dual configurations and regular graphs”, The Beauty of Geometry, Dover, ISBN0-486-40919-8

• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44,Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275

• Gropp, Harald (1990), “On the existence and non-existence of configurations nk", Journal of Combinatoricsand Information System Science 15: 34–48

• Gropp, Harald (1997), “Configurations and their realization”, Discrete Mathematics 174 (1–3): 137–151,doi:10.1016/S0012-365X(96)00327-5.

• Grünbaum, Branko (2006), “Configurations of points and lines”, in Davis, Chandler; Ellers, Erich W., TheCoxeter Legacy: Reflections and Projections, American Mathematical Society, pp. 179–225.

• Grünbaum, Branko (2009), Configurations of Points and Lines, Graduate Studies in Mathematics 103, Amer-ican Mathematical Society, ISBN 978-0-8218-4308-6.

• Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), Chelsea, pp. 94–170,ISBN 0-8284-1087-9.

• Kelly, L. M. (1986), “A resolution of the Sylvester–Gallai problem of J. P. Serre”, Discrete and ComputationalGeometry 1 (1): 101–104, doi:10.1007/BF02187687.

• Pisanski, Tomaž; Servatius, Brigitte (2013), Configurations from a Graphical Viewpoint, Springer, ISBN9780817683641.

8.10 External links• Weisstein, Eric W., “Configuration”, MathWorld.

Chapter 9

Content (measure theory)

In mathematics, a content is a real function µ defined on a field of sets A such that

1. µ(A) ∈ [0,∞] whenever A ∈ A.

2. µ(∅) = 0.

3. µ(A1 ∪A2) = µ(A1) + µ(A2) whenever A1, A2 ∈ A and A1 ∩A2 = ∅.

An example of a content is a measure, which is a σ-additive content defined on a σ-field. Every (real-valued) measureis a content, but not vice versa. Contents give a good notion of integrating bounded functions on a space but canbehave badly when integrating unbounded functions, while measures give a good notion of integrating unboundedfunctions.

9.1 Examples

An example of a content that is not a measure on a σ-algebra is the content on all subset of the positive integers thathas value 1/n on the integer n and is infinite on any infinite subset.An example of a content on the positive integers that is always finite but is not a measure can be given as follows. Takea positive linear functional on the bounded sequences that is 0 if the sequence has only a finite number of nonzeroelements and takes value 1 on the sequence 1, 1, 1, ...., so the functional in some sense gives an “average value” ofany bounded sequence. (Such a functional cannot be constructed explicitly, but exists by the Hahn-Banach theorem.)Then the content of a set of positive integers is the average value of the sequence that is 1 on this set and 0 elsewhere.Informally, one can think of the content of a subset of integers as the “chance” that a randomly chosen integer liesin this subset (though this is not compatible with the usual definitions of chance in probability theory, which assumecountable additivity).

9.2 Integration of bounded functions

In general integration of functions with respect to a content does not behave well. However there is a well-behavednotion of integration provided that the function is bounded and the total content of the space is finite, given as follows.Suppose that the total content of a space is finite. If f is a bounded function on the space such that the inverse imageof any open subset of the reals has a content, then we can define the integral of f with respect to the content as

∫fdλ = lim

n∑i=1

f(αi)λ(f−1(Ai))

where the Ai form a finite collections of disjoint half-open sets whose union covers the range of f, and αi is anyelement of Ai, and where the limit is taken as the diameters of the sets Ai tend to 0.

42

9.3. DUALS OF SPACES OF BOUNDED FUNCTIONS 43

9.3 Duals of spaces of bounded functions

Suppose that μ is a measure on some space X. The bounded measurable functions on X form a Banach space withrespect to the supremum norm. The positive elements of the dual of this space correspond to bounded contents λον Χ, with the value of λ on f given by the integral ∫fdλ. Similarly one can form the space of essentially boundedfunctions, with the norm given by the essential supremum, and the positive elements of the dual of this space aregiven by bounded contents that vanish on sets of measure 0.

9.4 Construction of a measure from a content

There are several ways to construct a measure μ from a content λ on a topological space. This section gives one suchmethod for locally compact Hausdorff spaces such that the content is defined on all compact subsets. In general themeasure is not an extension of the content, as the content may fail to be countably additive, and the measure mayeven be identically zero even if the content is not.First restrict the content to compact sets. This gives a function λ of compact sets C with the following properties:

1. λ(C) ∈ [0,∞] for all compact sets C

2. λ(∅) = 0.

3. λ(C1) ≤ λ(C2) whenever C1 ⊂ C2

4. λ(C1 ∪ C2) ≤ λ(C1) + λ(C2) for all pairs of compact sets

5. λ(C1 ∪ C2) = λ(C1) + λ(C2) for all pairs of disjoint compact sets.

There are also examples of functions λ as above not constructed from contents. An example is given by the construc-tion of Haar measure on a locally compact group. One method of constructing such a Haar measure is to produce aleft-invariant function λ as above on the compact subsets of the group, which can then be extended to a left-invariantmeasure.

9.4.1 Definition on open sets

Given λ as above, we define a function μ on all open sets by

µ(U) = supC⊂U

λ(C)

This has the following properties:

1. µ(U) ∈ [0,∞]

2. µ(∅) = 0.

3. µ(U1) ≤ µ(U2) whenever U1 ⊂ U2

4. µ(∪

n Un) ≤ ⊕nλ(Un) for any collection of open sets.

5. µ(∪

n Un) = ⊕nλ(Un) for any collection of disjoint open sets

9.4.2 Definition on all sets

Given μ as above, we extend the function μ to all subsets of the topological space by

µ(A) = infA⊂U

µ(U)

This is an outer measure, in other words it has the following properties:

44 CHAPTER 9. CONTENT (MEASURE THEORY)

1. µ(A) ∈ [0,∞]

2. µ(∅) = 0.

3. µ(A1) ≤ µ(A2) whenever A1 ⊂ A2

4. µ(∪

n An) ≤ ⊕nλ(An) for any countable collection of sets.

9.4.3 Construction of a measure

The function μ above is an outer measure on the family of all subsets. Therefore it becomes a measure when restrictedto the measurable subsets for the outer measure, which are the subsets E such that μ(X) = μ(X∩E) + μ(X\E) for allsubsets X. If the space is locally compact then every open set is measurable for this measure.The measure μ does not necessarily coincide with the content λ on compact sets, However it does if λ is regular inthe sense that for any compact C, λ(C) is the inf of λ(D) for compact sets D containing C in their interiors.

9.5 References• Halmos, Paul (1950), Measure Theory, Van Nostrand and Co.

• Mayrhofer, Karl (1952), Inhalt und Mass (Content and measure), Springer-Verlag, MR 0053185

Chapter 10

Dedekind number

The free distributive lattices of monotonic Boolean functions on 0, 1, 2, and 3 arguments, with 2, 3, 6, and 20 ele-ments respectively (move mouse over right diagram to see description)

In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind,who defined them in 1897. The Dedekind number M(n) counts the number of monotonic Boolean functions of nvariables. Equivalently, it counts the number of antichains of subsets of an n-element set, the number of elements ina free distributive lattice with n generators, or the number of abstract simplicial complexes with n elements.Accurate asymptotic estimates ofM(n)[1] and an exact expression as a summation,[2] are known. HoweverDedekind’sproblem of computing the values of M(n) remains difficult: no closed-form expression for M(n) is known, and exactvalues of M(n) have been found only for n ≤ 8.[3]

10.1 Definitions

A Boolean function is a function that takes as input n Boolean variables (that is, values that can be either false ortrue, or equivalently binary values that can be either 0 or 1), and produces as output another Boolean variable. Itis monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause theoutput to switch from false to true and not from true to false. The Dedekind number M(n) is the number of differentmonotonic Boolean functions on n variables.An antichain of sets (also known as a Sperner family) is a family of sets, none of which is contained in any other set.If V is a set of n Boolean variables, an antichain A of subsets of V defines a monotone Boolean function f, wherethe value of f is true for a given set of inputs if some subset of the true inputs to f belongs to A and false otherwise.Conversely every monotone Boolean function defines in this way an antichain, of the minimal subsets of Booleanvariables that can force the function value to be true. Therefore, the Dedekind number M(n) equals the number ofdifferent antichains of subsets of an n-element set.[4]

A third, equivalent way of describing the same class of objects uses lattice theory. From any two monotone Booleanfunctions f and g we can find two other monotone Boolean functions f ∧ g and f ∨ g, their logical conjunctionand logical disjunction respectively. The family of all monotone Boolean functions on n inputs, together with thesetwo operations, forms a distributive lattice, the lattice given by Birkhoff’s representation theorem from the partiallyordered set of subsets of the n variables with set inclusion as the partial order. This construction produces the freedistributive lattice with n generators.[5] Thus, the Dedekind numbers count the number of elements in free distributivelattices.[6]

The Dedekind numbers also count the number of abstract simplicial complexes on n elements, families of sets withthe property that any subset of a set in the family also belongs to the family. Any antichain determines a simplicialcomplex, the family of subsets of antichain members, and conversely the maximal simplices in a complex form anantichain.[7]

45

46 CHAPTER 10. DEDEKIND NUMBER

10.2 Example

For n = 2, there are six monotonic Boolean functions and six antichains of subsets of the two-element set x,y:

• The function f(x,y) = false that ignores its input values and always returns false corresponds to the emptyantichain Ø.

• The logical conjunction f(x,y) = x ∧ y corresponds to the antichain x,y containing the single set x,y.

• The function f(x,y) = x that ignores its second argument and returns the first argument corresponds to theantichain x containing the single set x

• The function f(x,y) = y that ignores its first argument and returns the second argument corresponds to theantichain y containing the single set y

• The logical disjunction f(x,y) = x ∨ y corresponds to the antichain x, y containing the two sets x andy.

• The function f(x,y) = true that ignores its input values and always returns true corresponds to the antichain Øcontaining only the empty set.

10.3 Values

The exact values of the Dedekind numbers are known for 0 ≤ n ≤ 8:

2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (sequence A000372 inOEIS).

The first six of these numbers are given by Church (1940). M(6) was calculated by Ward (1946),M(7) was calculatedby Church (1965) and Berman & Köhler (1976), and M(8) by Wiedemann (1991).If n is even, then M(n) must also be even.[8] The calculation of the fifth Dedekind number M(5) = 7581 disproved aconjecture by Garrett Birkhoff that M(n) is always divisible by (2n − 1)(2n − 2).[9]

10.4 Summation formula

Kisielewicz (1988) rewrote the logical definition of antichains into the following arithmetic formula for the Dedekindnumbers:

M(n) =22

n∑k=1

2n−1∏j=1

j−1∏i=0

1− bki bkj

log2 i∏m=0

(1− bim + bimbjm)

,

where bki is the i th bit of the number k , which can be written using the floor function as

bki =

⌊k

2i

⌋− 2

⌊k

2i+1

⌋.

However, this formula is not helpful for computing the values of M(n) for large n due to the large number of termsin the summation.

10.5. ASYMPTOTICS 47

10.5 Asymptotics

The logarithm of the Dedekind numbers can be estimated accurately via the bounds

(n

⌊n/2⌋

)≤ log2 M(n) ≤

(n

⌊n/2⌋

)(1 +O

( lognn

)).

Here the left inequality counts the number of antichains in which each set has exactly ⌊n/2⌋ elements, and the rightinequality was proven by Kleitman & Markowsky (1975).Korshunov (1981) provided the even more accurate estimates[10]

M(n) = (1 + o(1))2(n

⌊n/2⌋) exp a(n)

for even n, and

M(n) = (1 + o(1))2(n

⌊n/2⌋+1) exp(b(n) + c(n))

for odd n, where

a(n) =

(n

n/2− 1

)(2−n/2 + n22−n−5 − n2−n−4),

b(n) =

(n

(n− 3)/2

)(2−(n+3)/2 + n22−n−6 − n2−n−3),

and

c(n) =

(n

(n− 1)/2

)(2−(n+1)/2 + n22−n−4).

The main idea behind these estimates is that, in most antichains, all the sets have sizes that are very close to n/2.[10]

For n = 2, 4, 6, 8 Korshunov’s formula provides an estimate that is inaccurate by a factor of 9.8%, 10.2%, 4.1%, and−3.3%, respectively.[11]

10.6 Notes[1] Kleitman & Markowsky (1975); Korshunov (1981); Kahn (2002).

[2] Kisielewicz (1988).

[3] Wiedemann (1991).

[4] Kahn (2002).

[5] The definition of free distributive lattices used here allows as lattice operations any finite meet and join, including theempty meet and empty join. For the free distributive lattice in which only pairwise meets and joins are allowed, one shouldeliminate the top and bottom lattice elements and subtract two from the Dedekind numbers.

[6] Church (1940); Church (1965); Zaguia (1993).

[7] Kisielewicz (1988).

[8] Yamamoto (1953).

[9] Church (1940).

[10] Zaguia (1993).

[11] Brown, K. S., Generating the monotone Boolean functions

48 CHAPTER 10. DEDEKIND NUMBER

10.7 References• Berman, Joel; Köhler, Peter (1976), “Cardinalities of finite distributive lattices”, Mitt. Math. Sem. Giessen121: 103–124, MR 0485609.

• Church, Randolph (1940), “Numerical analysis of certain free distributive structures”, Duke MathematicalJournal 6: 732–734, doi:10.1215/s0012-7094-40-00655-x, MR 0002842.

• Church, Randolph (1965), “Enumeration by rank of the free distributive lattice with 7 generators”, Notices ofthe American Mathematical Society 11: 724. As cited by Wiedemann (1991).

• Dedekind, Richard (1897), "Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler”, Gesam-melte Werke 2, pp. 103–148.

• Kahn, Jeff (2002), “Entropy, independent sets and antichains: a new approach to Dedekind’s problem”, Pro-ceedings of the American Mathematical Society 130 (2): 371–378, doi:10.1090/S0002-9939-01-06058-0, MR1862115.

• Kisielewicz, Andrzej (1988), “A solution of Dedekind’s problem on the number of isotone Boolean functions”,Journal für die Reine und AngewandteMathematik 386: 139–144, doi:10.1515/crll.1988.386.139, MR 936995

• Kleitman, D.; Markowsky, G. (1975), “On Dedekind’s problem: the number of isotone Boolean functions. II”,Transactions of the American Mathematical Society 213: 373–390, doi:10.2307/1998052, MR 0382107.

• Korshunov, A. D. (1981), “The number of monotone Boolean functions”, Problemy Kibernet. 38: 5–108, MR0640855.

• Ward, Morgan (1946), “Note on the order of free distributive lattices”, Bulletin of the American MathematicalSociety 52: 423, doi:10.1090/S0002-9904-1946-08568-7.

• Wiedemann, Doug (1991), “A computation of the eighth Dedekind number”,Order 8 (1): 5–6, doi:10.1007/BF00385808,MR 1129608.

• Yamamoto, Koichi (1953), “Note on the order of free distributive lattices”, Science Reports of the KanazawaUniversity 2 (1): 5–6, MR 0070608.

• Zaguia, Nejib (1993), “Isotone maps: enumeration and structure”, in Sauer, N. W.; Woodrow, R. E.; Sands, B.,Finite and Infinite Combinatorics in Sets and Logic (Proc. NATO Advanced Study Inst., Banff, Alberta, Canada,May 4, 1991), Kluwer Academic Publishers, pp. 421–430, MR 1261220.

Chapter 11

Delta set

In mathematics, a delta set (or Δ-set) S is a combinatorial object that is useful in the construction and triangulation oftopological spaces, and also in the computation of related algebraic invariants of such spaces. A delta set is somewhatmore general than a simplicial complex, yet not quite as general as a simplicial set.

11.1 Definition and related data

Formally, a Δ-set is a sequence of sets Sn∞n=0 together with maps

di : Sn+1 → Sn

with i = 0,1,...,n + 1 for n ≥ 1 that satisfy

di dj = dj−1 di

whenever i < j.This definition generalizes the notion of a simplicial complex, where the Sn are the sets of n-simplices, and the diare the face maps. It is not as general as a simplicial set, since it lacks “degeneracies.”Given ∆ -sets S and T, a map of ∆ -sets is a collection

fn : Sn → Tn∞n=0

such that

fn di = di fn+1

whenever both sides of the equation are defined. With this notion, we can define the category of Δ-sets, whoseobjects are ∆ -sets and whose morphisms are maps of ∆ -sets.Each ∆ -set has a corresponding geometric realization, defined as

|S| =

( ∞⨿n=0

Sn ×∆n

)/∼

where we declare that

(σ, dit) ∼ (diσ, t) all for σ ∈ Sn, t ∈ ∆n−1.

49

50 CHAPTER 11. DELTA SET

Here, ∆n denotes the standard n-simplex, and

di : ∆n−1 → ∆n

is the inclusion of the i-th face. The geometric realization is a topological space with the quotient topology.The geometric realization of a ∆ -set S has a natural filtration

|S|0 ⊂ |S|1 ⊂ · · · ⊂ |S|,

where

|S|N =

(N⨿

n=0

Sn ×∆n

)/∼

is a “restricted” geometric realization.

11.2 Related functors

The geometric realization of a Δ-set described above defines a covariant functor from the category of Δ-sets to thecategory of topological spaces. Geometric realization takes a Δ-set to a topological space, and carries maps of Δ-setsto induced continuous maps between geometric realizations (which are topological spaces).If S is a Δ-set, there is an associated free abelian chain complex, denoted (ZS, ∂) , whose n-th group is the freeabelian group

(ZS)n = Z⟨Sn⟩,

generated by the set Sn , and whose n-th differential is defined by

∂n = d0 − d1 + d2 − · · ·+ (−1)ndn.

This defines a covariant functor from the category of Δ-sets to the category of chain complexes of abelian groups.A Δ-set is carried to the chain complex just described, and a map of Δ-sets is carried to a map of chain complexes,which is defined by extending the map of Δ-sets in the standard way using the universal property of free Abeliangroups.Given any topological spaceX, one can construct a Δ-set sing(X) as follows. A singular n-simplex inX is a continuousmap

σ : ∆n → X.

Define

singn(X)

to be the collection of all singular n-simplicies in X, and define

di : singi+1(X)→ singi(X)

by

11.3. AN EXAMPLE 51

di(σ) = σ di,

where again di is the i-th face map. One can check that this is in fact a Δ-set. This defines a covariant functor fromthe category of topological spaces to the category of Δ-sets. A topological space is carried to the Δ-set just described,and a continuous map of spaces is carried to a map of Δ-sets, which is given by composing the map with the singularn-simplices.

11.3 An example

This example illustrates the constructions described above. We can create a ∆ -set S whose geometric realization isthe unit circle S1 , and use it to compute the homology of this space. Thinking of S1 as an interval with the endpointsidentified, define

S0 = v, S1 = e,

with Sn = ∅ for all n ≥ 2. The only possible maps d0, d1 : S1 → S0, are

d0(e) = d1(e) = v.

It is simple to check that this is a ∆ -set, and that |S| ∼= S1 . Now, the associated chain complex (ZS, ∂) is

0 −→ Z⟨e⟩ ∂1−→Z⟨v⟩ −→ 0,

where

∂1(e) = d0(e)− d1(e) = v − v = 0.

In fact, ∂n = 0 for all n. The homology of this chain complex is also simple to compute:

H0(ZS) =ker ∂0im∂1

= Z⟨v⟩ ∼= Z,

H1(ZS) =ker ∂1im∂2

= Z⟨e⟩ ∼= Z.

All other homology groups are clearly trivial.One advantage of using ∆ -sets in this way is that the resulting chain complex is generally much simpler than thesingular chain complex. For reasonably simple spaces, all of the groups will be finitely generated, whereas the singularchain groups are, in general, not even countably generated.One drawback of this method is that one must prove that the geometric realization of the∆ -set is actually homeomorphicto the topological space in question. This can become a computational challenge as the∆ -set increases in complexity.

11.4 See also

• Simplicial complexes

• Simplicial sets

• Singular homology

52 CHAPTER 11. DELTA SET

11.5 References• Friedman, Greg (15 July 2008). “An elementary illustrated introduction to simplicial sets”. arXiv:0809.4221.

• Ranicki, Andrew A. (1993). Algebraic L-theory and Topological Manifolds (PDF). Cambridge Tracts in Math-ematics 102. Cambridge Univ. Press. ISBN 0-521-42024-5.

• Ranicki, Andrew; Weiss, Michael. “On the algebraic $L$-theory of Δ-sets”. arXiv:math.AT/0701833.

• Rourke, C. P.; Sanderson, B. J. (1971). "Δ-Sets I: Homotopy Theory”. The Quarterly Journal of Mathematics22 (3): 321–338. doi:10.1093/qmath/22.3.321.

Chapter 12

Delta-ring

In mathematics, a nonempty collection of setsR is called a δ-ring (pronounced delta-ring) if it is closed under union,relative complementation, and countable intersection:

1. A ∪B ∈ R if A,B ∈ R

2. A−B ∈ R if A,B ∈ R

3.∩∞

n=1 An ∈ R if An ∈ R for all n ∈ N

If only the first two properties are satisfied, then R is a ring but not a δ-ring. Every σ-ring is a δ-ring, but not everyδ-ring is a σ-ring.δ-rings can be used instead of σ-fields in the development of measure theory if one does not wish to allow sets ofinfinite measure.

12.1 See also• Ring of sets

• Sigma field

• Sigma ring

12.2 References• Cortzen, Allan. “Delta-Ring.” From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein.

http://mathworld.wolfram.com/Delta-Ring.html

53

Chapter 13

Dendroidal set

In mathematics, a dendroidal set is a generalization of simplicial sets introduced by Moerdijk & Weiss (2007). Theyhave the same relation to (colored symmetric) operads, also called symmetric multicategories, that simplicial sets haveto categories.

13.1 Definition

A dendroidal set is a contravariant functor from Ω to sets, where Ω is the tree category consisting of finite rootedtrees considered as operads, whose morphisms are operad morphisms. The trees are allowed to have some edges witha vertex on only one side; these are called outer edges, and the root is one of the outer edges.

13.2 References• Moerdijk, Ieke; Weiss, Ittay (2007), “Dendroidal sets”, Algebraic & Geometric Topology 7: 1441–1470,

doi:10.2140/agt.2007.7.1441, ISSN 1472-2747, MR 2366165

• Dendroidal set in nLab

54

Chapter 14

Discrete differential geometry

Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead ofsmooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computergraphics and topological combinatorics.

14.1 See also• Discrete Laplace operator

• Discrete exterior calculus

• Discrete Morse theory

• Topological combinatorics

• Spectral shape analysis

• Abstract differential geometry

• Analysis on fractals

14.2 References• Discrete differential geometry Forum

• Alexander I. Bobenko, Peter Schröder, John M. Sullivan, Günter M. Ziegler (2008). Discrete differentialgeometry. Birkhäuser Verlag AG. ISBN 978-3-7643-8620-7.

• Alexander I. Bobenko, Yuri B. Suris (2008), “Discrete Differential Geometry”, American Mathematical So-ciety, ISBN 978-0-8218-4700-8

55

Chapter 15

Disjoint sets

This article is about the mathematical concept. For the data structure, see Disjoint-set data structure.In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are

A BTwo disjoint sets.

sets whose intersection is the empty set.[1] For example, 1, 2, 3 and 4, 5, 6 are disjoint sets, while 1, 2, 3 and3, 4, 5 are not.

15.1 Generalizations

This definition of disjoint sets can be extended to any family of sets. A family of sets is pairwise disjoint ormutuallydisjoint if every two different sets in the family are disjoint.[1] For example, the collection of sets 1, 2, 3, ... is pairwise disjoint.Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two infinite setswhose intersection is a finite set may be said to be almost disjoint.[2]

In topology, there are various notions of separated sets with more strict conditions than disjointness. For instance,two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods. Similarly, ina metric space, positively separated sets are sets separated by a nonzero distance.[3]

56

15.2. EXAMPLES 57

A

BC

A pairwise disjoint family of sets

15.2 Examples

• The set of the drum and the guitar is disjoint to the set of the card and the book

• A pairwise disjoint family of sets

• A non pairwise disjoint family of sets

15.3 Intersections

Disjointness of two sets, or of a family of sets, may be expressed in terms of their intersections.

58 CHAPTER 15. DISJOINT SETS

Two sets A and B are disjoint if and only if their intersection A∩B is the empty set.[1] It follows from this definitionthat every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.[4]

A family F of sets is pairwise disjoint if, for every two sets in the family, their intersection is empty.[1] If the familycontains more than one set, this implies that the intersection of the whole family is also empty. However, a familyof only one set is pairwise disjoint, regardless of whether that set is empty, and may have a non-empty intersection.Additionally, a family of sets may have an empty intersection without being pairwise disjoint.[5] For instance, thethree sets 1, 2, 2, 3, 1, 3 have an empty intersection but are not pairwise disjoint. In fact, there are no twodisjoint sets in this collection. Also the empty family of sets is pairwise disjoint.[6]

A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that arepairwise disjoint. For instance, the closed intervals of the real numbers form a Helly family: if a family of closedintervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it mustbe pairwise disjoint.[7]

15.4 Disjoint unions and partitions

A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X.[8] Every partition canequivalently be described by an equivalence relation, a binary relation that describes whether two elements belongto the same set in the partition.[8] Disjoint-set data structures[9] and partition refinement[10] are two techniques incomputer science for efficiently maintaining partitions of a set subject to, respectively, union operations that mergetwo sets or refinement operations that split one set into two.A disjoint union may mean one of two things. Most simply, it may mean the union of sets that are disjoint.[11] Butif two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make themdisjoint before forming the union of the modified sets.[12] For instance two sets may be made disjoint by replacingeach element by an ordered pair of the element and a binary value indicating whether it belongs to the first or secondset.[13] For families of more than two sets, one may similarly replace each element by an ordered pair of the elementand the index of the set that contains it.[14]

15.5 See also• Hyperplane separation theorem for disjoint convex sets

• Mutually exclusive events

• Relatively prime, numbers with disjoint sets of prime divisors

• Set packing, the problem of finding the largest disjoint subfamily of a family of sets

15.6 References[1] Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 15, ISBN 9780387900926.

[2] Halbeisen, Lorenz J. (2011), Combinatorial Set Theory: With a Gentle Introduction to Forcing, Springer monographs inmathematics, Springer, p. 184, ISBN 9781447121732.

[3] Copson, Edward Thomas (1988), Metric Spaces, Cambridge Tracts in Mathematics 57, Cambridge University Press, p. 62,ISBN 9780521357326.

[4] Oberste-Vorth, Ralph W.; Mouzakitis, Aristides; Lawrence, Bonita A. (2012), Bridge to Abstract Mathematics, MAAtextbooks, Mathematical Association of America, p. 59, ISBN 9780883857793.

[5] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2010), A Transition to Advanced Mathematics, Cengage Learning,p. 95, ISBN 9780495562023.

[6] See answers to the question ″Is the empty family of sets pairwise disjoint?″

[7] Bollobás, Béla (1986), Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, Cam-bridge University Press, p. 82, ISBN 9780521337038.

15.7. EXTERNAL LINKS 59

[8] Halmos (1960), p. 28.

[9] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), “Chapter 21: Data structures forDisjoint Sets”, Introduction to Algorithms (Second ed.), MIT Press, pp. 498–524, ISBN 0-262-03293-7.

[10] Paige, Robert; Tarjan, Robert E. (1987), “Three partition refinement algorithms”, SIAM Journal on Computing 16 (6):973–989, doi:10.1137/0216062, MR 917035.

[11] Ferland, Kevin (2008), Discrete Mathematics: An Introduction to Proofs and Combinatorics, Cengage Learning, p. 45,ISBN 9780618415380.

[12] Arbib, Michael A.; Kfoury, A. J.; Moll, Robert N. (1981), A Basis for Theoretical Computer Science, The AKM series inTheoretical Computer Science: Texts and monographs in computer science, Springer-Verlag, p. 9, ISBN 9783540905738.

[13] Monin, Jean François; Hinchey, Michael Gerard (2003),Understanding FormalMethods, Springer, p. 21, ISBN 9781852332471.

[14] Lee, John M. (2010), Introduction to Topological Manifolds, Graduate Texts in Mathematics 202 (2nd ed.), Springer, p.64, ISBN 9781441979407.

15.7 External links• Weisstein, Eric W., “Disjoint Sets”, MathWorld.

Chapter 16

Dold–Kan correspondence

In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence states[1] that thereis an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicialabelian groups. Moreover, under the equivalence, the homology group of a chain complex is the homotopy group ofthe corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, thecorrespondence preserves the respective standard model structures.)Example: Let C be a chain complex that has an abelian group A in degree n and zero in other degrees. Then thecorresponding simplicial group is the Eilenberg–MacLane space K(A,n) .There is also an ∞-category-version of a Dold–Kan correspondence.[2]

16.1 References[1] Goerss–Jardine 1999, Ch 3. Corollary 2.3

[2] Lurie 2012, § 1.2.4.

• Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress in Mathematics 174. Basel, Boston,Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.

• J. Lurie, Higher Algebra, last updated August 2012

• A. Mathew, The Dold-Kan correspondence

16.2 Further reading• J. Lurie, DAG-I

16.3 External links• Dold-Kan correspondence in nLab

60

Chapter 17

Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set Ω satisfying a setof axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems (Dynkin himselfused this term) or d-system.[1] These set families have applications in measure theory and probability.The primary relevance of λ-systems are their use in applications of the π-λ theorem.

17.1 Definitions

Let Ω be a nonempty set, and let D be a collection of subsets of Ω (i.e., D is a subset of the power set of Ω). ThenD is a Dynkin system if

1. Ω ∈ D ,

2. if A, B ∈ D and A ⊆ B, then B \ A ∈ D ,

3. if A1, A2, A3, ... is a sequence of subsets in D and An ⊆ An₊₁ for all n ≥ 1, then∪∞

n=1 An ∈ D .

Equivalently, D is a Dynkin system if

1. Ω ∈ D ,

2. if A ∈ D, then Ac ∈ D,

3. if A1, A2, A3, ... is a sequence of subsets in D such that Ai ∩ Aj = Ø for all i ≠ j, then∪∞

n=1 An ∈ D .

The second definition is generally preferred as it usually is easier to check.An important fact is that a Dynkin system which is also a π-system (i.e., closed under finite intersection) is a σ-algebra.This can be verified by noting that condition 3 and closure under finite intersection implies closure under countableunions.Given any collection J of subsets of Ω , there exists a unique Dynkin system denoted DJ which is minimal withrespect to containing J . That is, if D is any Dynkin system containing J , then DJ ⊆ D . DJ is called theDynkin system generated by J . Note D∅ = ∅,Ω . For another example, let Ω = 1, 2, 3, 4 and J = 1 ;then DJ = ∅, 1, 2, 3, 4,Ω .

17.2 Dynkin’s π-λ theorem

If P is a π-system and D is a Dynkin system with P ⊆ D , then σP ⊆ D . In other words, the σ-algebra generatedby P is contained in D .One application of Dynkin’s π-λ theorem is the uniqueness of a measure that evaluates the length of an interval(known as the Lebesgue measure):

61

62 CHAPTER 17. DYNKIN SYSTEM

Let (Ω, B, λ) be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let μ be another measure on Ωsatisfying μ[(a,b)] = b − a, and letD be the family of sets S such that μ[S] = λ[S]. Let I = (a,b),[a,b),(a,b],[a,b] : 0 <a ≤ b < 1 , and observe that I is closed under finite intersections, that I ⊂ D, and that B is the σ-algebra generated byI. It may be shown that D satisfies the above conditions for a Dynkin-system. From Dynkin’s π-λ Theorem it followsthat D in fact includes all of B, which is equivalent to showing that the Lebesgue measure is unique on B.Additional applications are in the article on π-systems.

17.3 Notes[1] Charalambos Aliprantis, Kim C. Border (2006). Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd ed. Springer.

Retrieved August 23, 2010.

17.4 References• Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN

0-387-22833-0.

• Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2.

• David Williams (2007). Probability with Martingales. Cambridge University Press. p. 193. ISBN 0-521-40605-6.

This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Chapter 18

Erdős–Ko–Rado theorem

In combinatorics, the Erdős–Ko–Rado theorem of Paul Erdős, Chao Ko, and Richard Rado is a theorem onintersecting set families. It is part of the theory of hypergraphs, specifically, uniform hypergraphs of rank r.The theorem is as follows. If n ≥ 2r and A is a family of distinct subsets of 1, 2, ..., n such that each subset is ofsize r and each pair of subsets intersects, then the maximum number of sets that can be in A is given by the binomialcoefficient

(n− 1

r − 1

).

(Since a family of sets may be called a hypergraph, and since every set in A has size r, A is a uniform hypergraph ofrank r.)According to Erdős (1987) the theorem was proved in 1938, but was not published until 1961 in an apparently moregeneral form. The subsets in question were only required to be size at most r , and with the additional requirementthat no subset be contained in any other. This statement is not actually more general: any subset that has size lessthan r can be increased to size r to make the above statement apply.

18.1 Proof

The original proof of 1961 used induction on n. In 1972, Gyula O. H. Katona gave the following short proof usingdouble counting.Suppose we have some such family of subsetsA. Arrange the elements of 1, 2, ..., n in any cyclic order, and considerthe sets from A that form intervals of length r within this cyclic order. For example if n = 8 and r = 3, we couldarrange the numbers 1, 2, ..., 8 into the cyclic order (3,1,5,4,2,7,6,8), which has eight intervals:

(3,1,5), (1,5,4), (5,4,2), (4,2,7), (2,7,6), (7,6,8), (6,8,3), and (8,3,1).

However, it is not possible for all of the intervals of the cyclic order to belong to A, because some pairs of themare disjoint. Katona’s key observation is that at most r of the intervals for a single cyclic order may belong to A. Tosee this, note that if (a1, a2, ..., ar) is one of these intervals in A, then every other interval of the same cyclic orderthat belongs to A separates ai and ai ₊ ₁ for some i (that is, it contains precisely one of these two elements). Thetwo intervals that separate these elements are disjoint, so at most one of them can belong to A. Thus, the number ofintervals in A is one plus the number of separated pairs, which is at most (r-1).Based on this idea, we may count the number of pairs (S,C), where S is a set in A and C is a cyclic order for which Sis an interval, in two ways. First, for each set S one may generate C by choosing one of r! permutations of S and (n− r)! permutations of the remaining elements, showing that the number of pairs is |A|r!(n − r)!. And second, thereare (n − 1)! cyclic orders, each of which has at most r intervals of A, so the number of pairs is at most r(n − 1)!.Combining these two counts gives the inequality

63

64 CHAPTER 18. ERDŐS–KO–RADO THEOREM

|A|r!(n− r)! ≤ r(n− 1)!

and dividing both sides by r!(n − r)! gives the result

|A| ≤ r(n− 1)!

r!(n− r)!=

(n− 1

r − 1

).

Two constructions for an intersecting family of r-sets: fix one element and choose the remaining elements in all possible ways, or(when n = 2r) exclude one element and choose all subsets of the remaining elements. Here n = 4 and r = 2.

18.2 Families of maximum size

There are two different and straightforward constructions for an intersecting family of r-element sets achievingthe Erdős–Ko–Rado bound on cardinality. First, choose any fixed element x, and let A consist of all r-subsets of1, 2, ..., n that include x. For instance, if n = 4, r = 2, and x = 1, this produces the family of three 2-sets

1,2, 1,3, 1,4.

Any two sets in this family intersect, because they both include x. Second, when n = 2r and with x as above, let Aconsist of all r-subsets of 1, 2, ..., n that do not include x. For the same parameters as above, this produces thefamily

2,3, 2,4, 3,4.

Any two sets in this family have a total of 2r = n elements among them, chosen from the n − 1 elements that areunequal to x, so by the pigeonhole principle they must have an element in common.When n > 2r, families of the first type (variously known as sunflowers, stars, dictatorships, centred families, principalfamilies) are the unique maximum families. Friedgut (2008) proved that in this case, a family which is almost ofmaximum size has an element which is common to almost all of its sets. This property is known as stability.

18.3 Maximal intersecting families

An intersecting family of r-element sets may be maximal, in that no further set can be added without destroying theintersection property, but not of maximum size. An example with n = 7 and r = 3 is the set of 7 lines of the Fanoplane, much less than the Erdős–Ko–Rado bound of 15.

18.4. REFERENCES 65

The seven points and seven lines (one drawn as a circle) of the Fano plane form a maximal intersecting family.

18.4 References• Erdős, P. (1987), “My joint work with Richard Rado”, in Whitehead, C., Surveys in combinatorics, 1987:Invited Papers for the Eleventh British Combinatorial Conference (PDF), London Mathematical Society LectureNote Series 123, Cambridge University Press, pp. 53–80, ISBN 978-0-521-34805-8.

• Erdős, P.; Ko, C.; Rado, R. (1961), “Intersection theorems for systems of finite sets” (PDF), The QuarterlyJournal of Mathematics. Oxford. Second Series 12: 313–320, doi:10.1093/qmath/12.1.313.

• Friedgut, Ehud (2008), “On the measure of intersecting families, uniqueness and stability” (PDF), Combina-torica 28 (5): 503–528, doi:10.1007/s00493-008-2318-9

• Katona, G. O. H. (1972), “A simple proof of the Erdös-Chao Ko-Rado theorem”, Journal of CombinatorialTheory, Series B 13 (2): 183–184, doi:10.1016/0095-8956(72)90054-8.

Chapter 19

Family of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family ofsubsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.The term “collection” is used here because, in some contexts, a family of sets may be allowed to contain repeatedcopies of any given member,[1][2][3] and in other contexts it may form a proper class rather than a set.

19.1 Examples• The power set P(S) is a family of sets over S.

• The k-subsets S(k) of a set S form a family of sets.

• Let S = a,b,c,1,2, an example of a family of sets over S (in the multiset sense) is given by F = A1, A2, A3,A4 where A1 = a,b,c, A2 = 1,2, A3 = 1,2 and A4 = a,b,1.

• The class Ord of all ordinal numbers is a large family of sets; that is, it is not itself a set but instead a properclass.

19.2 Special types of set family• A Sperner family is a family of sets in which none of the sets contains any of the others. Sperner’s theorem

bounds the maximum size of a Sperner family.

• A Helly family is a family of sets such that any minimal subfamily with empty intersection has bounded size.Helly’s theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families.

19.3 Properties• Any family of subsets of S is itself a subset of the power set P(S) if it has no repeated members.

• Any family of sets without repetitions is a subclass of the proper class V of all sets (the universe).

• Hall’s marriage theorem, due to Philip Hall gives necessary and sufficient conditions for a finite family ofnon-empty sets (repetitions allowed) to have a system of distinct representatives.

19.4 Related concepts

Certain types of objects from other areas of mathematics are equivalent to families of sets, in that they can be describedpurely as a collection of sets of objects of some type:

66

19.5. SEE ALSO 67

• A hypergraph, also called a set system, is formed by a set of vertices together with another set of hyperedges,each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any familyof sets can be interpreted as a hypergraph that has the union of the sets as its vertices.

• An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shapeformed by unions of line segments, triangles, tetrahedra, and higher-dimensional simplices, joined face to face.In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family offinite sets without repetitions in which the subsets of any set in the family also belong to the family forms anabstract simplicial complex.

• An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called theincidence relation, specifying which points belong to which lines. An incidence structure can be specified bya family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to eachline, and any family of sets can be interpreted as an incidence structure in this way.

• A binary block code consists of a set of codewords, each of which is a string of 0s and 1s, all the same length.When each pair of codewords has large Hamming distance, it can be used as an error-correcting code. A blockcode can also be described as a family of sets, by describing each codeword as the set of positions at which itcontains a 1.

19.5 See also• Indexed family

• Class (set theory)

• Combinatorial design

• Russell’s paradox (or Set of sets that do not contain themselves)

19.6 Notes[1] Brualdi 2010, pg. 322

[2] Roberts & Tesman 2009, pg. 692

[3] Biggs 1985, pg. 89

19.7 References• Biggs, Norman L. (1985), Discrete Mathematics, Oxford: Clarendon Press, ISBN 0-19-853252-0

• Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Upper Saddle River, NJ: Prentice Hall, ISBN0-13-602040-2

• Roberts, Fred S.; Tesman, Barry (2009), Applied Combinatorics (2nd ed.), Boca Raton: CRC Press, ISBN978-1-4200-9982-9

Chapter 20

Field of sets

“Set algebra” redirects here. For the basic properties and laws of sets, see Algebra of sets.

In mathematics a field of sets is a pair ⟨X,F⟩ where X is a set andF is an algebra over X i.e., a non-empty subsetof the power set of X closed under the intersection and union of pairs of sets and under complements of individualsets. In other words F forms a subalgebra of the power set Boolean algebra of X . (Many authors refer to F itselfas a field of sets. The word “field” in “field of sets” is not used with the meaning of field from field theory.) Elementsof X are called points and those of F are called complexes and are said to be the admissible sets of X .Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can berepresented as a field of sets.

20.1 Fields of sets in the representation theory of Boolean algebras

20.1.1 Stone representation

Every finite Boolean algebra can be represented as a whole power set - the power set of its set of atoms; each elementof the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power setrepresentation can be constructed more generally for any complete atomic Boolean algebra.In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representationby considering fields of sets instead of whole power sets. To do this we first observe that the atoms of a finite Booleanalgebra correspond to its ultrafilters and that an atom is below an element of a finite Boolean algebra if and only ifthat element is contained in the ultrafilter corresponding to the atom. This leads us to construct a representation of aBoolean algebra by taking its set of ultrafilters and forming complexes by associating with each element of the Booleanalgebra the set of ultrafilters containing that element. This construction does indeed produce a representation of theBoolean algebra as a field of sets and is known as the Stone representation. It is the basis of Stone’s representationtheorem for Boolean algebras and an example of a completion procedure in order theory based on ideals or filters,similar to Dedekind cuts.Alternatively one can consider the set of homomorphisms onto the two element Boolean algebra and form complexesby associating each element of the Boolean algebra with the set of such homomorphisms that map it to the topelement. (The approach is equivalent as the ultrafilters of a Boolean algebra are precisely the pre-images of the topelements under these homomorphisms.) With this approach one sees that Stone representation can also be regardedas a generalization of the representation of finite Boolean algebras by truth tables.

20.1.2 Separative and compact fields of sets: towards Stone duality

• A field of sets is called separative (or differentiated) if and only if for every pair of distinct points there is acomplex containing one and not the other.

• A field of sets is called compact if and only if for every proper filter overX the intersection of all the complexes

68

20.2. FIELDS OF SETS WITH ADDITIONAL STRUCTURE 69

contained in the filter is non-empty.

These definitions arise from considering the topology generated by the complexes of a field of sets. Given a field ofsets X = ⟨X,F⟩ the complexes form a base for a topology, we denote the corresponding topological space by T (X). Then

• T (X) is always a zero-dimensional space.

• T (X) is a Hausdorff space if and only if X is separative.

• T (X) is a compact space with compact open sets F if and only if X is compact.

• T (X) is a Boolean space with clopen sets F if and only if X is both separative and compact (in which case itis described as being descriptive)

The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space isknown as the Stone space of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexesof the Stone representation. The area of mathematics known as Stone duality is founded on the fact that the Stonerepresentation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a dualityexists between Boolean algebras and Boolean spaces.

20.2 Fields of sets with additional structure

20.2.1 Sigma algebras and measure spaces

If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebraand the corresponding field of sets is called a measurable space. The complexes of a measurable space are calledmeasurable sets.A measure space is a triple ⟨X,F , µ⟩ where ⟨X,F⟩ is a measurable space and µ is a measure defined on it. If µis in fact a probability measure we speak of a probability space and call its underlying measurable space a samplespace. The points of a sample space are called samples and represent potential outcomes while the measurable sets(complexes) are called events and represent properties of outcomes for which we wish to assign probabilities. (Manyuse the term sample space simply for the underlying set of a probability space, particularly in the case where everysubset is an event.) Measure spaces and probability spaces play a foundational role in measure theory and probabilitytheory respectively.The Loomis-Sikorski theorem provides a Stone-type duality between abstract sigma algebras and measurable spaces.

20.2.2 Topological fields of sets

A topological field of sets is a triple ⟨X, T ,F⟩ where ⟨X, T ⟩ is a topological space and ⟨X,F⟩ is a field of setswhich is closed under the closure operator of T or equivalently under the interior operator i.e. the closure and interiorof every complex is also a complex. In other words F forms a subalgebra of the power set interior algebra on ⟨X, T ⟩.Every interior algebra can be represented as a topological field of sets with its interior and closure operators corre-sponding to those of the topological space.Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interiorand closure. The Stone representation of a Boolean algebra can be regarded as such a topological field of sets.

Algebraic fields of sets and Stone fields

A topological field of sets is called algebraic if and only if there is a base for its topology consisting of complexes.If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets areprecisely the open complexes. Moreover the open complexes form a base for the topology.

70 CHAPTER 20. FIELD OF SETS

Topological fields of sets that are separative, compact and algebraic are calledStone fields and provide a generalizationof the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation ofits underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated bythe complexes corresponding to the open elements of the interior algebra (which form a base for a topology). Thesecomplexes are then precisely the open complexes and the construction produces a Stone field representing the interioralgebra - the Stone representation.

20.2.3 Preorder fields

A preorder field is a triple ⟨X,≤,F⟩ where ⟨X,≤⟩ is a preordered set and ⟨X,F⟩ is a field of sets.Like the topological fields of sets, preorder fields play an important role in the representation theory of interior alge-bras. Every interior algebra can be represented as a preorder field with its interior and closure operators correspondingto those of the Alexandrov topology induced by the preorder. In other words

Int(S) = x ∈ X : there exists a y ∈ S with y ≤ x andCl(S) = x ∈ X : there exists a y ∈ S with x ≤ y for all S ∈ F

Preorder fields arise naturally in modal logic where the points represent the possible worlds in the Kripke semanticsof a theory in the modal logic S4 (a formal mathematical abstraction of epistemic logic), the preorder represents theaccessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worldsin which individual sentences in the theory hold, providing a representation of the Lindenbaum-Tarski algebra of thetheory.

Algebraic and canonical preorder fields

A preorder field is called algebraic if and only if it has a set of complexes A which determines the preorder in thefollowing manner: x ≤ y if and only if for every complex S ∈ A , x ∈ S implies y ∈ S . The preorder fieldsobtained from S4 theories are always algebraic, the complexes determining the preorder being the sets of possibleworlds in which the sentences of the theory closed under necessity hold.A separative compact algebraic preorder field is said to be canonical. Given an interior algebra, by replacing thetopology of its Stone representation with the corresponding canonical preorder (specialization preorder) we obtain arepresentation of the interior algebra as a canonical preorder field. By replacing the preorder by its correspondingAlexandrov topology we obtain an alternative representation of the interior algebra as a topological field of sets. (Thetopology of this "Alexandrov representation" is just the Alexandrov bi-coreflection of the topology of the Stonerepresentation.)

20.2.4 Complex algebras and fields of sets on relational structures

The representation of interior algebras by preorder fields can be generalized to a representation theorem for arbi-trary (normal) Boolean algebras with operators. For this we consider structures ⟨X, (Ri)I ,F⟩ where ⟨X, (Ri)I⟩ is arelational structure i.e. a set with an indexed family of relations defined on it, and ⟨X,F⟩ is a field of sets. The com-plex algebra (or algebra of complexes) determined by a field of sets X = ⟨X, (Ri)I ,F⟩ on a relational structure,is the Boolean algebra with operators

C(X) = ⟨F ,∩,∪, ′, ∅, X, (fi)I⟩

where for all i ∈ I , if Ri is a relation of arity n+ 1 , then fi is an operator of arity n and for all S1, ..., Sn ∈ F

fi(S1, ..., Sn) = x ∈ X : there exist x1 ∈ S1, ..., xn ∈ Sn such that Ri(x1, ..., xn, x)

This construction can be generalized to fields of sets on arbitrary algebraic structures having both operators andrelations as operators can be viewed as a special case of relations. If F is the whole power set of X then C(X) iscalled a full complex algebra or power algebra.

20.3. SEE ALSO 71

Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in thesense that it is isomorphic to the complex algebra corresponding to the field.(Historically the term complex was first used in the case where the algebraic structure was a group and has its originsin 19th century group theory where a subset of a group was called a complex.)

20.3 See also• List of Boolean algebra topics

• Algebra of sets

• Sigma algebra

• Measure theory

• Probability theory

• Interior algebra

• Alexandrov topology

• Stone’s representation theorem for Boolean algebras

• Stone duality

• Boolean ring

• Preordered field

20.4 References• Goldblatt, R., Algebraic Polymodal Logic: A Survey, Logic Journal of the IGPL, Volume 8, Issue 4, p. 393-450,

July 2000

• Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic, 44, p. 173-242, 1989

• Johnstone, Peter T. (1982). Stone spaces (3rd ed.). Cambridge: Cambridge University Press. ISBN 0-521-33779-8.

• Naturman, C.A., Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Math-ematics, 1991

• Patrick Blackburn, Johan F.A.K. van Benthem, Frank Wolter ed., Handbook of Modal Logic, Volume 3 ofStudies in Logic and Practical Reasoning, Elsevier, 2006

20.5 External links• Hazewinkel, Michiel, ed. (2001), “Algebra of sets”, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

Chapter 21

Finite character

In mathematics, a family F of sets is of finite character provided it has the following properties:

1. For each A ∈ F , every finite subset of A belongs to F .

2. If every finite subset of a given set A belongs to F , then A belongs to F .

21.1 Properties

A family F of sets of finite character enjoys the following properties:

1. For each A ∈ F , every (finite or infinite) subset of A belongs to F .

2. Tukey’s lemma: In F , partially ordered by inclusion, the union of every chain of elements of F also belong toF , therefore, by Zorn’s lemma, F contains at least one maximal element.

21.2 Example

Let V be a vector space, and let F be the family of linearly independent subsets of V. Then F is a family of finitecharacter (because a subset X ⊆ V is linearly dependent iff X has a finite subset which is linearly dependent). There-fore, in every vector space, there exists a maximal family of linearly independent elements. As a maximal family isa vector basis, every vector space has a (possibly infinite) vector basis.This article incorporates material from finite character on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

72

Chapter 22

Finite intersection property

In general topology, a branch of mathematics, a collectionA of subsets of a setX is said to have the finite intersectionproperty (FIP) if the intersection over any finite subcollection of A is nonempty. It has the strong finite intersectionproperty (SFIP) if the intersection over any finite subcollection of A is infinite.A centered system of sets is a collection of sets with the finite intersection property.

22.1 Definition

Let X be a set with A = Aii∈I a family of subsets of X . Then the collection A has the finite intersection property(FIP), if any finite subcollection J ⊆ I has non-empty intersection

∩i∈J Ai.

22.2 Discussion

Clearly the empty set cannot belong to any collection with the finite intersection property. The condition is triviallysatisfied if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), andit is also trivially satisfied if the collection is nested, meaning that the collection is totally ordered by inclusion (equiv-alently, for any finite subcollection, a particular element of the subcollection is contained in all the other elements ofthe subcollection), e.g. the nested sequence of intervals (0, 1/n). These are not the only possibilities however. Forexample, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion withdigit 0 in the i'th decimal place, then any finite intersection is nonempty (just take 0 in those finitely many places and1 in the rest), but the intersection of all Xi for i ≥ 1 is empty, since no element of (0, 1) has all zero digits.The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact ifand only if every collection of closed sets satisfying the finite intersection property has nonempty intersection itself.[1]

This formulation of compactness is used in some proofs of Tychonoff’s theorem and the uncountability of the realnumbers (see next section)

22.3 Applications

Theorem. Let X be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open.Then X is uncountable.Proof. We will show that if U ⊆ X is nonempty and open, and if x is a point of X, then there is a neighbourhoodV ⊂ U whose closure doesn’t contain x (x may or may not be in U). Choose y in U different from x (if x is in U,then there must exist such a y for otherwise U would be an open one point set; if x isn’t in U, this is possible sinceU is nonempty). Then by the Hausdorff condition, choose disjoint neighbourhoods W and K of x and y respectively.Then K ∩ U will be a neighbourhood of y contained in U whose closure doesn’t contain x as desired.

Now suppose f : N → X is a bijection, and let xi : i ∈ N denote the image of f. Let X be the first open set

73

74 CHAPTER 22. FINITE INTERSECTION PROPERTY

and choose a neighbourhood U1 ⊂ X whose closure doesn’t contain x1. Secondly, choose a neighbourhood U2 ⊂U1 whose closure doesn’t contain x2. Continue this process whereby choosing a neighbourhood Un₊₁ ⊂ Un whoseclosure doesn’t contain xn₊₁. Then the collection Ui : i ∈ N satisfies the finite intersection property and hence theintersection of their closures is nonempty (by the compactness of X). Therefore there is a point x in this intersection.No xi can belong to this intersection because xi doesn’t belong to the closure of Ui. This means that x is not equal toxi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.All the conditions in the statement of the theorem are necessary:1. We cannot eliminate the Hausdorff condition; a countable set with the indiscrete topology is compact, has morethan one point, and satisfies the property that no one point sets are open, but is not uncountable.2. We cannot eliminate the compactness condition as the set of all rational numbers shows.3. We cannot eliminate the condition that one point sets cannot be open as a finite space given the discrete topologyshows.Corollary. Every closed interval [a, b] with a < b is uncountable. Therefore, R is uncountable.Corollary. Every perfect, locally compact Hausdorff space is uncountable.Proof. Let X be a perfect, compact, Hausdorff space, then the theorem immediately implies that X is uncountable.If X is a perfect, locally compact Hausdorff space which is not compact, then the one-point compactification of X isa perfect, compact Hausdorff space. Therefore the one point compactification of X is uncountable. Since removinga point from an uncountable set still leaves an uncountable set, X is uncountable as well.

22.4 Examples

A filter has the finite intersection property by definition.

22.5 Theorems

Let X be nonempty, F ⊆ 2X, F having the finite intersection property. Then there exists an F′ ultrafilter (in 2X) suchthat F ⊆ F′.See details and proof in Csirmaz & Hajnal (1994).[2] This result is known as ultrafilter lemma.

22.6 Variants

A family of sets A has the strong finite intersection property (sfip), if every finite subfamily of A has infiniteintersection.

22.7 References[1] A space is compact iff any family of closed sets having fip has non-empty intersection at PlanetMath.org.

[2] Csirmaz, László; Hajnal, András (1994), Matematikai logika (In Hungarian), Budapest: Eötvös Loránd University.

• Finite intersection property at PlanetMath.org.

Chapter 23

Fisher’s inequality

Fisher’s inequality, is a necessary condition for the existence of a balanced incomplete block design which satisfiescertain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population geneticist andstatistician, was concerned with the design of experiments studying the differences among several different varietiesof plants, under each of a number of different growing conditions, called “blocks”.Let:

• v be the number of varieties of plants;

• b be the number of blocks.

It was required that:

• k different varieties are in each block, k < v; no variety occurs twice in any one block;

• any two varieties occur together in exactly λ blocks;

• each variety occurs in exactly r blocks.

Fisher’s inequality states simply that

b ≥ v.

23.1 Proof

Let the incidence matrix M be a v×b matrix defined so that Mi,j is 1 if element i is in block j and 0 otherwise. ThenB=MMT is a v×v matrix such that Bi,i = r and Bi,j = λ for i ≠ j. Since r ≠ λ, det(B) ≠ 0, so rank(B) = v; on the otherhand, rank(B) = rank(M) ≤ b, so v ≤ b.

23.2 Generalization

Fisher’s inequality is valid for more general classes of designs. A “pairwise balanced design” (or PBD) is a set Xtogether with a family of subsets of X (which need not have the same size and may contain repeats) such that everypair of distinct elements of X is contained in exactly λ (a positive integer) subsets. The set X is allowed to be one ofthe subsets, and if all the subsets are copies of X, the PBD is called “trivial”. The size of X is v and the number ofsubsets in the family (counted with multiplicity) is b.Theorem: For any non-trivial PBD, v ≤ b.[1]

75

76 CHAPTER 23. FISHER’S INEQUALITY

Ronald Fisher

This result also generalizes the Erdős-De Bruijn theorem:For a PBD with λ = 1 having no blocks of size 1 or size v, v ≤ b, with equality if the PBD is a projective plane or anear-pencil (meaning that exactly n - 1 of the points are collinear).[2]

23.3 Notes

[1] Stinson 2003, pg.193

[2] Stinson 2003, pg.183

23.4. REFERENCES 77

23.4 References• R. C. Bose, “A Note on Fisher’s Inequality for Balanced Incomplete Block Designs”, Annals of MathematicalStatistics, 1949, pages 619–620.

• R. A. Fisher, “An examination of the different possible solutions of a problem in incomplete blocks”, Annalsof Eugenics, volume 10, 1940, pages 52–75.

• Stinson, Douglas R. (2003), Combinatorial Designs: Constructions and Analysis, New York: Springer, ISBN0-387-95487-2

• Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P.[Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3.

Chapter 24

Generalized quadrangle

GQ(2,2), the Doily

In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yetcontaining many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are thegeneralized n-gons and near n-gons with n = 4. They are also precisely the partial geometries pg(s,t,α) with α = 1.

78

24.1. DEFINITION 79

24.1 Definition

A generalized quadrangle is an incidence structure (P,B,I), with I ⊆ P × B an incidence relation, satisfying certainaxioms. Elements of P are by definition the points of the generalized quadrangle, elements of B the lines. The axiomsare the following:

• There is an s (s ≥ 1) such that on every line there are exactly s + 1 points. There is at most one point on twodistinct lines.

• There is a t (t ≥ 1) such that through every point there are exactly t + 1 lines. There is at most one line throughtwo distinct points.

• For every point p not on a line L, there is a unique line M and a unique point q, such that p is on M, and q onM and L.

(s,t) are the parameters of the generalized quadrangle. The parameters are allowed to be infinite. If either s or t isone, the generalized quadrangle is called trivial. For example, the 3x3 grid with P = 1,2,3,4,5,6,7,8,9 and L =123, 456, 789, 147, 258, 369 is a trivial GQ with s = 2 and t = 1. A generalized quadrangle with parameters (s,t)is often denoted by GQ(s,t).The smallest non-trivial generalized quadrangle is GQ(2,2), whose representation has been dubbed “the doily” byStan Payne in 1973.

24.2 Properties

• |P | = (st+ 1)(s+ 1)

• |B| = (st+ 1)(t+ 1)

• (s+ t)|st(s+ 1)(t+ 1)

• s = 1 =⇒ t ≤ s2

• t = 1 =⇒ s ≤ t2

24.3 Graphs

There are two interesting graphs that can be obtained from a generalized quadrangle.

• The collinearity graph having as vertices the points of a generalized quadrangle, with the collinear pointsconnected. This graph is a strongly regular graph with parameters ((s+1)(st+1), s(t+1), s-1, t+1) where (s,t) isthe order of the GQ.

• The incidence graph whose vertices are the points and lines of the generalized quadrangle and two vertices areadjacent if one is a point, the other a line and the point lies on the line. The incidence graph of a generalizedquadrangle is characterized by being a connected, bipartite graph with diameter four and girth eight. Thereforeit is an example of a Cage. Incidence graphs of configurations are today generally called Levi graphs, but theoriginal Levi graph was the incidence graph of the GQ(2,2).

24.4 Duality

If (P,B,I) is a generalized quadrangle with parameters (s,t), then (B,P,I−1), with I−1 the inverse incidence relation, isalso a generalized quadrangle. This is the dual generalized quadrangle. Its parameters are (t,s). Even if s = t, the dualstructure need not be isomorphic with the original structure.

80 CHAPTER 24. GENERALIZED QUADRANGLE

Line graph of generalized quadrangle GQ(2,4)

24.5 Generalized quadrangles with lines of size 3

There are precisely five (possible degenerate) generalized quadrangles where each line has three point incident with it,the quadrangle with empty line set, the quadrangle with all lines through a fixed point corresponding to the windmillgraph Wd(3,n), grid of size 3x3, the W(2) quadrangle and the unique GQ(2,4). These five quadrangles correspondsto the five root systems in the ADE classes An, Dn, E6, E7 and E8 , i.e., the simply laced root systems. See [1] and.[2]

24.6 Classical generalized quadrangles

When looking at the different cases for polar spaces of rank at least three, and extrapolating them to rank 2, one findsthese (finite) generalized quadrangles :

• A hyperbolic quadric Q+(3, q) , a parabolic quadric Q(4, q) and an elliptic quadric Q−(5, q) are the onlypossible quadrics in projective spaces over finite fields with projective index 1. We find these parametersrespectively :

Q(3, q) : s = q, t = 1

24.7. NON-CLASSICAL EXAMPLES 81

Q(4, q) : s = q, t = q

Q(5, q) : s = q, t = q2

• A hermitian variety H(n, q2) has projective index 1 if and only if n is 3 or 4. We find :

H(3, q2) : s = q2, t = q

H(4, q2) : s = q2, t = q3

• A symplectic polarity in PG(2d+1, q) has a maximal isotropic subspace of dimension 1 if and only if d = 1. Here, we find a generalized quadrangle W (3, q) , with s = q, t = q .

The generalized quadrangle derived from Q(4, q) is always isomorphic with the dual of W (3, q) , and they are bothself-dual and thus isomorphic to each other if and only if q is even.

24.7 Non-classical examples• Let O be a hyperoval in PG(2, q) with q an even prime power, and embed that projective (desarguesian) plane

π into PG(3, q) . Now consider the incidence structure T ∗2 (O) where the points are all points not in π ,

the lines are those not on π , intersecting π in a point of O, and the incidence is the natural one. This is a(q-1,q+1)-generalized quadrangle.

• Let q be a prime power (odd or even) and consider a symplectic polarity θ in PG(3, q) . Choose a randompoint p and define π = pθ . Let the lines of our incidence structure be all absolute lines not on π together withall lines through p which are not on π , and let the points be all points of PG(3, q) except those in π . Theincidence is again the natural one. We obtain once again a (q-1,q+1)-generalized quadrangle

24.8 Restrictions on parameters

By using grids and dual grids, any integer z, z ≥ 1 allows generalized quadrangles with parameters (1,z) and (z,1).Apart from that, only the following parameters have been found possible until now, with q an arbitrary prime power :

(q, q)

(q, q2) and (q2, q)

(q2, q3) and (q3, q2)

(q − 1, q + 1) and (q + 1, q − 1)

24.9 References[1] Cameron P.J.; Goethals, J.M.; Seidel, J.J; Shult, E. E. Line graphs, root systems and elliptic geometry

[2] http://www.win.tue.nl/~aeb/2WF02/genq.pdf

• S. E. Payne and J. A. Thas. Finite generalized quadrangles. Research Notes in Mathematics, 110. Pitman(Advanced Publishing Program), Boston, MA, 1984. vi+312 pp. ISBN 0-273-08655-3, link http://cage.ugent.be/~bamberg/FGQ.pdf

Chapter 25

Greedoid

In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originallyintroduced by Whitney in 1935 to study planar graphs and was later used by Edmonds to characterize a class of opti-mization problems that can be solved by greedy algorithms. Around 1980, Korte and Lovász introduced the greedoidto further generalize this characterization of greedy algorithms; hence the name greedoid. Besides mathematicaloptimization, greedoids have also been connected to graph theory, language theory, poset theory, and other areas ofmathematics.

25.1 Definitions

A set system (F, E) is a collection F of subsets of a ground set E (i.e. F is a subset of the power set of E). Whenconsidering a greedoid, a member of F is called a feasible set. When considering a matroid, a feasible set is alsoknown as an independent set.An accessible set system (F, E) is a set system in which every nonempty feasible set X contains an element x suchthat X\x is feasible. This implies that any nonempty, finite accessible set system necessarily contains the empty set∅.A greedoid (F, E) is an accessible set system that satisfies the exchange property:

• for all X,Y ∈ F with |X| > |Y|, there is some x ∈ X\Y such that Y∪x ∈ F

(Note: Some people reserve the term exchange property for a condition on the bases of a greedoid, and prefer to callthe above condition the “Augmentation Property”.)A basis of a greedoid is a maximal feasible set, meaning it is a feasible set but not contained in any other one. Abasis of a subset X of E is a maximal feasible set contained in X.The rank of a greedoid is the size of a basis. By the exchange property, all bases have the same size. Thus, the rankfunction is well defined. The rank of a subset X of E is the size of a basis of X.

25.2 Classes of greedoids

Most classes of greedoids have many equivalent definitions in terms of set system, language, poset, simplicial complex,and so on. The following description takes the traditional route of listing only a couple of the more well-knowncharacterizations.An interval greedoid (F, E) is a greedoid that satisfies the Interval Property:

• if A, B, C ∈ F with A ⊆ B ⊆ C, then, for all x ∈ E\C, (A∪x ∈ F and C∪x ∈ F) implies B∪x ∈ F

Equivalently, an interval greedoid is a greedoid such that the union of any two feasible sets is feasible if it is containedin another feasible set.

82

25.3. EXAMPLES 83

An antimatroid (F, E) is a greedoid that satisfies the Interval Property without Upper Bounds:

• if A, B ∈ F with A ⊆ B, then, for all x ∈ E\B, A∪x ∈ F implies B∪x ∈ F

Equivalently, an antimatroid is (i) a greedoid with a unique basis; or (ii) an accessible set system closed under union.It is easy to see that an antimatroid is also an interval greedoid.A matroid (F, E) is a greedoid that satisfies the Interval Property without Lower Bounds:

• if B, C ∈ F with B ⊆ C, then, for all x ∈ E\C, C∪x ∈ F implies B∪x ∈ F

It is easy to see that a matroid is also an interval greedoid.

25.3 Examples

• Consider an undirected graph G. Let the ground set be the edges of G and the feasible sets be the edge set ofeach forest (i.e. subgraph containing no cycle) of G. This set system is called the cycle matroid. A set systemis said to be a graphic matroid if it is the cycle matroid of some graph. (Originally cycle matroid was definedon circuits, or minimal dependent sets. Hence the name cycle.)

• Consider a finite, undirected graph G rooted at the vertex r. Let the ground set be the vertices of G and thefeasible sets be the vertex subsets containing r that induce connected subgraphs of G. This is called the vertexsearch greedoid and is a kind of antimatroid.

• Consider a finite, directed graph D rooted at r. Let the ground set be the (directed) edges of D and the feasiblesets be the edge sets of each directed subtree rooted at r with all edges pointing away from r. This is called theline search greedoid, or directed branching greedoid. It is an interval greedoid, but neither an antimatroidnor a matroid.

• Consider an m-by-n matrix M. Let the ground set E be the indices of the columns from 1 to n and the feasiblesets be F = X ⊆ E: submatrix M₁,...,|X|,X is an invertible matrix. This is called the Gaussian eliminationgreedoid because this structure underlies the Gaussian elimination algorithm. It is a greedoid, but not aninterval greedoid.

25.4 Greedy algorithm

In general, a greedy algorithm is just an iterative process in which a locally best choice, usually an input of minimumweight, is chosen each round until all available choices have been exhausted. In order to describe a greedoid-basedcondition in which a greedy algorithm is optimal, we need some more common terminologies in greedoid theory.Without loss of generality, we consider a greedoid G = (F, E) with E finite.A subset X of E is rank feasible if the largest intersection of X with any feasible set has size equal to the rank of X.In a matroid, every subset of E is rank feasible. But the equality does not hold for greedoids in general.A function w: E → ℝ is R-compatible if x ∈ E: w(x) ≥ c is rank feasible for all real numbers c.An objective function f: 2S → ℝ is linear over a set S if, for all X ⊆ S, we have f(X) = Σₓ ∈ X w(x) for some weightfunction w: S → ℜ.Proposition. A greedy algorithm is optimal for every R-compatible linear objective function over a greedoid.The intuition behind this proposition is that, during the iterative process, each optimal exchange of minimum weightis made possible by the exchange property, and optimal results are obtainable from the feasible sets in the underlyinggreedoid. This result guarantees the optimality of many well-known algorithms. For example, a minimum spanningtree of a weighted graph may be obtained using Kruskal’s algorithm, which is a greedy algorithm for the cycle matroid.

84 CHAPTER 25. GREEDOID

25.5 See also• Matroid

• Polymatroid

25.6 References• Björner, Anders; Ziegler, Günter M. (1992), White, Neil, ed., “Matroid Applications”, Matroid applications,

Encyclopedia of Mathematics and its Applications (Cambridge: Cambridge University Press) 40: 284–357,doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537, Zbl 0772.05026 |chapter= ig-nored (help)

• Edmonds, Jack (1971), “Matroids and the greedy algorithm”,Mathematical Programming 1: 127–113, doi:10.1007/BF01584082,Zbl 0253.90027.

• Helman, Paul; Moret, Bernard M. E.; Shapiro, Henry D. (1993), “An exact characterization of greedy struc-tures”, SIAM Journal on Discrete Mathematics 6 (2): 274–283, doi:10.1137/0406021, Zbl 0798.68061.

• Korte, Bernhard; Lovász, László (1981), “Mathematical structures underlying greedy algorithms”, in Gecseg,Ferenc, Fundamentals of Computation Theory: Proceedings of the 1981 International FCT-Conference, Szeged,Hungaria, August 24–28, 1981, Lecture Notes in Computer Science 117, Berlin: Springer-Verlag, pp. 205–209, doi:10.1007/3-540-10854-8_22, Zbl 0473.68019.

• Korte, Bernhard; Lovász, László; Schrader, Rainer (1991), Greedoids, Algorithms and Combinatorics 4, NewYork, Berlin: Springer-Verlag, ISBN 3-540-18190-3, Zbl 0733.05023.

• Oxley, James G. (1992), Matroid theory, Oxford Science Publications, Oxford: Oxford University Press, ISBN0-19-853563-5, Zbl 0784.05002.

• Whitney, Hassler (1935), “On the abstract properties of linear independence”, American Journal of Mathe-matics 57 (3): 509–533, doi:10.2307/2371182, JSTOR 2371182, Zbl 0012.00404.

25.7 External links• Introduction to Greedoids

• Theory of Greedy Algorithms

• Submodular Functions and Optimization

• Matchings, Matroids and Submodular Functions

Chapter 26

Helly family

In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an emptyintersection has k or fewer sets in it. Equivalently, every finite subfamily such that every k -fold intersection is non-empty has non-empty total intersection.[1]

The k-Helly property is the property of being a Helly family of order k.[2] These concepts are named after EduardHelly (1884 - 1943); Helly’s theorem on convex sets, which gave rise to this notion, states that convex sets in Euclideanspace of dimension n are a Helly family of order n + 1.[1] The number k is frequently omitted from these names inthe case that k = 2.

26.1 Examples

• In the family of all subsets of the set a,b,c,d, the subfamily a,b,c, a,b,d, a,c,d, b,c,d has an emptyintersection, but removing any set from this subfamily causes it to have a nonempty intersection. Therefore,it is a minimal subfamily with an empty intersection. It has four sets in it, and is the largest possible minimalsubfamily with an empty intersection, so the family of all subsets of the set a,b,c,d is a Helly family of order4.

• Let I be a finite set of closed intervals of the real line with an empty intersection. Let A be the interval whose leftendpoint a is as large as possible, and let B be the interval whose right endpoint b is as small as possible. Then,if a were less than or equal to b, all numbers in the range [a,b] would belong to all invervals of I, violatingthe assumption that the intersection of I is empty, so it must be the case that a > b. Thus, the two-intervalsubfamily A,B has an empty intersection, and the family I cannot be minimal unless I = A,B. Therefore,all minimal families of intervals with empty intersections have two or fewer intervals in them, showing that theset of all intervals is a Helly family of order 2.[3]

• The family of infinite arithmetic progressions of integers also has the 2-Helly property. That is, whenever afinite collection of progressions has the property that no two of them are disjoint, then there exists an integerthat belongs to all of them; this is the Chinese remainder theorem.[2]

26.2 Formal definition

More formally, a Helly family of order k is a set system (F, E), with F a collection of subsets of E, such that, forevery finite G ⊆ F with

∩X∈G

X = ∅,

we can find H ⊆ G such that

85

86 CHAPTER 26. HELLY FAMILY

∩X∈H

X = ∅

and

|H| ≤ k. [1]

In some cases, the same definition holds for every subcollection G, regardless of finiteness. However, this is a morerestrictive condition. For instance, the open intervals of the real line satisfy the Helly property for finite subcollections,but not for infinite subcollections: the intervals (0,1/i) (for i = 0, 1, 2, ...) have pairwise nonempty intersections, buthave an empty overall intersection.

26.3 Helly dimension

If a family of sets is a Helly family of order k, that family is said to have Helly number k. The Helly dimension ofa metric space is one less than the Helly number of the family of metric balls in that space; Helly’s theorem impliesthat the Helly dimension of a Euclidean space equals its dimension as a real vector space.[4]

The Helly dimension of a subset S of a Euclidean space, such as a polyhedron, is one less than the Helly number ofthe family of translates of S.[5] For instance, the Helly dimension of any hypercube is 1, even though such a shapemay belong to a Euclidean space of much higher dimension.[6]

Helly dimension has also been applied to other mathematical objects. For instance Domokos (2007) defines the Hellydimension of a group (an algebraic structure formed by an invertible and associative binary operation) to be one lessthan the Helly number of the family of left cosets of the group.[7]

26.4 The Helly property

If a family of nonempty sets has an empty intersection, its Helly number must be at least two, so the smallest k forwhich the k-Helly property is nontrivial is k = 2. The 2-Helly property is also known as theHelly property. A 2-Hellyfamily is also known as a Helly family.[1][2]

A convex metric space in which the closed balls have the 2-Helly property (that is, a space with Helly dimension 1, inthe stronger variant of Helly dimension for infinite subcollections) is called injective or hyperconvex.[8] The existenceof the tight span allows any metric space to be embedded isometrically into a space with Helly dimension 1.[9]

26.5 References[1] Bollobás, Béla (1986), Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, Cam-

bridge University Press, p. 82, ISBN 9780521337038.

[2] Duchet, Pierre (1995), “Hypergraphs”, in Graham, R. L.; Grötschel, M.; Lovász, L., Handbook of combinatorics, Vol. 1,2, Amsterdam: Elsevier, pp. 381–432, MR 1373663. See in particular Section 2.5, “Helly Property”, pp. 393–394.

[3] This is the one-dimensional case of Helly’s theorem. For essentially this proof, with a colorful phrasing involving sleep-ing students, see Savchev, Svetoslav; Andreescu, Titu (2003), “27 Helly’s Theorem for One Dimension”, MathematicalMiniatures, New Mathematical Library 43, Mathematical Association of America, pp. 104–106, ISBN 9780883856451.

[4] Martini, Horst (1997), Excursions Into Combinatorial Geometry, Springer, pp. 92–93, ISBN 9783540613411.

[5] Bezdek, Károly (2010), Classical Topics in Discrete Geometry, Springer, p. 27, ISBN 9781441906007.

[6] Sz.-Nagy, Béla (1954), “Ein Satz über Parallelverschiebungen konvexer Körper”, Acta Universitatis Szegediensis 15: 169–177, MR 0065942.

[7] Domokos, M. (2007), “Typical separating invariants”, TransformationGroups 12 (1): 49–63, arXiv:math/0511300, doi:10.1007/s00031-005-1131-4, MR 2308028.

26.5. REFERENCES 87

[8] Deza, Michel Marie; Deza, Elena (2012), Encyclopedia of Distances, Springer, p. 19, ISBN 9783642309588

[9] Isbell, J. R. (1964), “Six theorems about injective metric spaces”, Comment. Math. Helv. 39: 65–76, doi:10.1007/BF02566944.

Chapter 27

Helly’s theorem

Helly’s theorem for the Euclidean plane: if a family of convex sets has a nonempty intersection for every triple of sets, then the wholefamily has a nonempty intersection.

Helly’s theorem is a basic result in discrete geometry describing the ways that convex sets may intersect each other.It was discovered by Eduard Helly in 1913,[1] but not published by him until 1923, by which time alternative proofsby Radon (1921) and König (1922) had already appeared. Helly’s theorem gave rise to the notion of a Helly family.

27.1 Statement

Let X1, ..., Xn be a finite collection of convex subsets of Rd, with n > d. If the intersection of every d+1 of these setsis nonempty, then the whole collection has a nonempty intersection; that is,

n∩j=1

Xj = ∅.

For infinite collections one has to assume compactness:Let Xα be a collection of compact convex subsets of Rd, such that every subcollection of cardinality at most d+1has nonempty intersection, then the whole collection has nonempty intersection.

27.2 Proof

We prove the finite version, using Radon’s theorem as in the proof by Radon (1921). The infinite version then followsby the finite intersection property characterization of compactness: a collection of closed subsets of a compact space

88

27.3. SEE ALSO 89

has a non-empty intersection if and only if every finite subcollection has a non-empty intersection (once you fix asingle set, the intersection of all others with it are closed subsets of a fixed compact space).The proof is based on induction:Base Case: Let n = d+2. By our assumptions, for every j = 1, ..., n there is a point xj that is in the common intersectionof all Xi with the possible exception of Xj. Now we apply Radon’s theorem to the set A = x1, ..., xn, which furnishesus with disjoint subsets A1, A2 of A such that the convex hull of A1 intersects the convex hull of A2. Suppose that pis a point in the intersection of these two convex hulls. We claim that

p ∈n∩

j=1

Xj .

Indeed, consider any j ∈ 1, ..., n. We shall prove that p ∈ Xj. Note that the only element of A that may not be in Xjis xj. If xj ∈ A1, then xj ∉ A2, and therefore Xj ⊃ A2. Since Xj is convex, it then also contains the convex hull of A2

and therefore also p ∈ Xj. Likewise, if xj ∉ A1, then Xj ⊃ A1, and by the same reasoning p ∈ Xj. Since p is in everyXj, it must also be in the intersection.Above, we have assumed that the points x1, ..., xn are all distinct. If this is not the case, say xi = xk for some i ≠ k,then xi is in every one of the sets Xj, and again we conclude that the intersection is nonempty. This completes theproof in the case n = d+2.Inductive Step: Suppose n > d+1 and that the statement is true for n−1. The argument above shows that any sub-collection of d+2 sets will have nonempty intersection. We may then consider the collection where we replace thetwo sets Xn₋₁ and Xn with the single set Xn₋₁ ∩ Xn. In this new collection, every subcollection of d+1 sets will havenonempty intersection. The inductive hypothesis therefore applies, and shows that this new collection has nonemptyintersection. This implies the same for the original collection, and completes the proof.

27.3 See also

• Carathéodory’s theorem

• Shapley–Folkman lemma

• Krein–Milman theorem

• Choquet theory

• Radon’s theorem

27.4 Notes[1] Danzer, Grünbaum & Klee (1963).

27.5 References

• Bollobás, B. (2006), “Problem 29, Intersecting Convex Sets: Helly’s Theorem”, The Art ofMathematics: CoffeeTime in Memphis, Cambridge University Press, pp. 90–91, ISBN 0-521-69395-0.

• Danzer, L.; Grünbaum, B.; Klee, V. (1963), “Helly’s theorem and its relatives”, Convexity, Proc. Symp. PureMath. 7, American Mathematical Society, pp. 101–179.

• Eckhoff, J. (1993), “Helly, Radon, and Carathéodory type theorems”, Handbook of Convex Geometry A, B,Amsterdam: North-Holland, pp. 389–448.

• Heinrich Guggenheimer (1977) Applicable Geometry, page 137, Krieger, Huntington ISBN 0-88275-368-1 .

90 CHAPTER 27. HELLY’S THEOREM

• Helly, E. (1923), "Über Mengen konvexer Körper mit gemeinschaftlichen Punkten”, Jahresbericht der DeutschenMathematiker-Vereinigung 32: 175–176.

• König, D. (1922), "Über konvexe Körper”,Mathematische Zeitschrift 14 (1): 208–220, doi:10.1007/BF01215899.

• Radon, J. (1921), “Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten”, Mathematische An-nalen 83 (1–2): 113–115, doi:10.1007/BF01464231.

Chapter 28

Incidence structure

In mathematics, an abstract system consisting of two types of objects and a single relationship between these typesof objects is called an incidence structure. Consider the points and lines of the Euclidean plane as the two types ofobjects and ignore all the properties of this geometry except for the relation of which points are on which lines for allpoints and lines. What is left is the incidence structure of the Euclidean plane.Incidence structures are most often considered in the geometrical context where they are abstracted from, and hencegeneralize, planes (such as affine, projective, and Möbius planes), but the concept is very broad and not limited togeometric settings. Even in a geometric setting, incidence structures are not limited to just points and lines; higher-dimensional objects (planes, solids, n-spaces, conics, etc.) can be used. The study of finite structures is sometimescalled finite geometry.[1]

28.1 Formal definition and terminology

An incidence structure is a triple (P, L, I) where P is a set whose elements are called points, L is a disjoint set whoseelements are called lines and I ⊆ P × L is the incidence relation. The elements of I are called flags. If (p, l) is in Ithen it was typical to say that point p “lies on” line l or that the line l “passes through” point p. However, today a more“symmetric” terminology is preferred to reflect the symmetric nature of this relation, so one says that "p is incidentwith l" or that "l is incident with p" and uses the notation p I l in lieu of (p, l) ∈ I.[2]

In some common situations L may be a set of subsets of P in which case incidence I will be containment (p I l if andonly if p is a member of l). Incidence structures of this type are called set-theoretic.[3] This is not always the case, forexample, if P is a set of vectors and L a set of square matrices, we may define I = (v, M) : vector v is an eigenvectorof matrix M . This example also shows that while the geometric language of points and lines is used, the objecttypes need not be these geometric objects.

28.2 Examples

Main article: Incidence geometry

An incidence structure is uniform if each line is incident with the same number of points. Each of these examples,except the second, is uniform with three points per line.Any graph (need not be simple, loops and multiple edges are allowed) is a uniform incidence structure with two pointsper line. For these examples, the vertices of the graph form the point set, the edges of the graph form the line set,and incidence means that a vertex is an endpoint of an edge.Incidence structures are seldom studied in their full generality; it is typical to study incidence structures that satisfysome additional axioms. For instance, a partial linear space is an incidence structure that satisfies:

1. Any two distinct points are incident with at most one common line, and

2. Every line is incident with at least two points.

91

92 CHAPTER 28. INCIDENCE STRUCTURE

If the first axiom above is replaced by the stronger:

1. Any two distinct points are incident with exactly one common line,

the incidence structure is called a linear space.[4][5]

28.3 Dual structure

If we interchange the role of “points” and “lines” in

C = (P, L, I)

we obtain the dual structure,

C∗ = (L, P, I∗),

where I∗ is the inverse relation of I. It follows immediately from the definition that:

C∗∗ = C.

This is an abstract version of projective duality.[2]

A structure C that is isomorphic to its dual C∗ is called self-dual. The Fano plane above is a self-dual incidencestructure.

28.4 Other terminology

The concept of an incidence structure is very simple and has arisen in several disciplines, each introducing its ownvocabulary and specifying the types of questions that are typically asked about these structures. Incidence structuresuse a geometric terminology, but in graph theoretic terms they are called hypergraphs and in design theoretic termsthey are called block designs. They are also known as a set system or family of sets in a general context.

28.4.1 Hypergraphs

Main article: HypergraphEach hypergraph or set system can be regarded as an incidence structure in which the universal set plays the role of“points”, the corresponding family of sets plays the role of “lines” and the incidence relation is set membership "∈".Conversely, every incidence structure can be viewed as a hypergraph by identifying the lines with the sets of pointsthat are incident with them.

28.4.2 Block designs

Main article: Block design

A (general) block design is a set X together with a family of subsets (repeated subsets are allowed), F of X. As anincidence structure, X is the set of points, F the set of lines, usually called blocks in this context (repeated blocksmust have distinct names, so F is actually a set and not a multiset). If all the subsets in F have the same size, theblock design is called uniform. If each element of X appears in the same number of subsets, the block design is saidto be regular. The dual of a uniform design is a regular design and vice versa.

28.4. OTHER TERMINOLOGY 93

1

2

3

4

5

6

7

Seven points are elements of seven lines in the Fano plane

Example: Fano plane

Consider the block design/hypergraph given by:

P = 1, 2, 3, 4, 5, 6, 7

L = 1, 2, 3, 1, 4, 5, 1, 6, 7, 2, 4, 6, 2, 5, 7, 3, 4, 7, 3, 5, 6

This incidence structure is called the Fano plane. As a block design it is both uniform and regular.In the labeling given, the lines are precisely the subsets of the points that consist of three points whose labels add upto zero using nim addition. Alternatively, each number, when written in binary, can be identified with a non-zerovector of length three over the binary field. Three vectors that generate a subspace form a line; in this case, that isequivalent to their vector sum being the zero vector.

94 CHAPTER 28. INCIDENCE STRUCTURE

28.5 Representations

Incidence structures may be represented in many ways. If the sets P and L are finite these representations can com-pactly encode all the relevant information concerning the structure.

28.5.1 Incidence matrix

Main article: Incidence matrix

The incidence matrix of a (finite) incidence structure is a matrix with entries 0 or 1 that has its rows indexed by thepoints pᵢ and columns indexed by the lines l where the ij-th entry is a 1 if pᵢ I l and 0 otherwise.[6] An incidencematrix is not uniquely determined since it depends upon the arbitrary ordering of the points and the lines.[7]

The non-uniform incidence structure pictured above (#2 of the examples) is given by:

P = A, B, C, D, E, P L = l = C, P, E , m = P , n = P, D , o = P, A , p = A, B , q = P, B .

An incidence matrix for this structure is :

0 0 0 1 1 00 0 0 0 1 11 0 0 0 0 00 0 1 0 0 01 0 0 0 0 01 1 1 1 0 1

which corresponds to the incidence table:

If an incidence structure C has an incidence matrix M, then the dual structure C∗ has the transpose matrix MT as itsincidence matrix (and is defined by that matrix).An incidence structure is self-dual if there exists an ordering of the points and lines so that the incidence matrixconstructed with that ordering is a symmetric matrix.With the labels as given in example #1 above and with points ordered A, B, C, D, G, F, E and lines ordered l, p, n, s,r, m, q, the Fano plane has the incidence matrix:

1 1 1 0 0 0 01 0 0 1 1 0 01 0 0 0 0 1 10 1 0 1 0 1 00 1 0 0 1 0 10 0 1 1 0 0 10 0 1 0 1 1 0

.

Since this is a symmetric matrix, the Fano plane is a self-dual incidence structure.

28.5.2 Pictorial representations

An incidence figure (that is, a depiction of an incidence structure), is constructed by representing the points by dotsin a plane and having some visual means of joining the dots to correspond to lines.[7] The dots may be placed in anymanner, there are no restrictions on distances between points or any relationships between points. In an incidence

28.5. REPRESENTATIONS 95

structure there is no concept of a point being between two other points; the order of points on a line is undefined.Compare this with ordered geometry, which does have a notion of betweenness. The same statements can be madeabout the depictions of the lines. In particular, lines need not be depicted by “straight line segments” (see examples1, 3 and 4 above). As with the pictorial representation of graphs, the crossing of two “lines” at anyplace other than adot, has no meaning in terms of the incidence structure, it is only an accident of the representation. These incidencefigures may at times resemble graphs, but they aren't graphs unless the incidence structure is a graph.

Realizability

Incidence structures can be modelled by points and curves in the Euclidean plane with the usual geometric meaningof incidence. Some incidence structures admit representation by points and (straight) lines. Structures that can be arecalled realizable. If no ambient space is mentioned then the Euclidean plane is assumed. The Fano plane (#1 above)is not realizable since it needs at least one curve. The Möbius-Kantor configuration (#4 above) is not realizable inthe Euclidean plane, but it is realizable in the complex plane.[8] On the other hand, examples #2 and #5 above arerealizable and the incidence figures given there demonstrate this. Steinitz (1894)[9] has shown that n3-configurations(incidence structures with n points and n lines, three points per line and three lines through each point) are eitherrealizable or require the use of only one curved line in their representations.[10] The Fano plane is the unique (73) andthe Möbius-Kantor configuration is the unique (83).

28.5.3 Incidence graph (Levi graph)

1 1,3,2

1,7,6

1,5,4

2

2,7,5

3 3,7,4

3,5,6

7

6

5

4

2,6,4

Heawood graph with labeling

96 CHAPTER 28. INCIDENCE STRUCTURE

Each incidence structure C corresponds to a bipartite graph called the Levi graph or incidence graph of the structure.As any bipartite graph is two colorable, the Levi graph can be given a black and white vertex coloring, where blackvertices correspond to points and white vertices correspond to lines of C. The edges of this graph correspond to theflags (incident point/line pairs) of the incidence structure. The original Levi graph was the incidence graph of thegeneralized quadrangle of order two (example #3 above),[11] but the term has been extended by H.S.M. Coxeter[12]

to refer to an incidence graph of any incidence structure.[13]

Levi graph of the Möbius-Kantor configuration (#4)

Levi graph examples

The Levi graph of the Fano plane is the Heawood graph. Since the Heawood graph is connected and vertex-transitive,there exists an automorphism (such as the one defined by a reflection about the vertical axis in the figure of theHeawood graph) interchanging black and white vertices. This, in turn, implies that the Fano plane is self-dual.The specific representation, on the left, of the Levi graph of the Möbius-Kantor configuration (example #4 above)illustrates that a rotation of π/4 about the center (either clockwise or counterclockwise) of the diagram interchangesthe blue and red vertices and maps edges to edges. That is to say that there exists a color interchanging automorphismof this graph. Consequently, the incidence structure known as the Möbius-Kantor configuration is self-dual.

28.6. GENERALIZATION 97

28.6 Generalization

It is possible to generalize the notion of an incidence structure to include more than two types of objects. A structurewith k types of objects is called an incidence structure of rank k or a rank k geometry.[13] Formally these are definedas k + 1 tuples S = (P1, P2, ..., P , I) with Pᵢ ∩ P = ∅ and

I ⊆∪i<j

Pi × Pj .

The Levi graph for these structures is defined as a multipartite graph with vertices corresponding to each type beingcolored the same.

28.7 See also

• Binary relation

• Incidence (geometry)

• Incidence geometry

• Projective configuration

28.8 Notes

[1] Colbourn & Dinitz 2007, p. 702

[2] Dembowski 1968, pp. 1-2

[3] Biliotti, Jha & Johnson 2001, p. 508

[4] The term linear space is also used to refer to vector spaces, but this will rarely cause confusion.

[5] Moorhouse 2007, p. 5

[6] The other convention of indexing the rows by lines and the columns by points is also widely used.

[7] Beth, Jungnickel & Lenz 1986, p. 17

[8] Pisanski & Servatius 2014, p. 222

[9] E. Steinitz (1894), Über die Construction der Configurationen n3, Dissertation, Breslau

[10] Gropp, Harald (1997), “Configurations and their realizations”, Discrete Mathematics 174: 137–151

[11] Levi, F. W. (1942), Finite Geometrical Systems, Calcutta: University of Calcutta, MR 0006834

[12] Coxeter, H.S.M. (1950), “Self-dual configurations and regular graphs”, Bulletin of the American Mathematical Society 56:413–455

[13] Pisanski & Servatius 2014, p. 158

98 CHAPTER 28. INCIDENCE STRUCTURE

28.9 References• Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge University Press, ISBN

3-411-01675-2

• Biliotti, Mauro; Jha, Vikram; Johnson, Norman L. (2001), Foundations of Translation Planes, Marcel Dekker,ISBN 0-8247-0609-9

• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, ISBN 1-58488-506-8

• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44,Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275

• Incidence Geometry (2007) by Eric Moorhouse

• Pisanski, Tomaž; Servatius, Brigitte (2013),Configurations from aGraphical Viewpoint, Springer, doi:10.1007/978-0-8176-8364-1, ISBN 978-0-8176-8363-4

28.10 Further reading• CRC Press (2000). Handbook of discrete and combinatorial mathematics, (Chapter 12.2), ISBN 0-8493-0149-

1

• Harold L. Dorwart (1966) The Geometry of Incidence, Prentice Hall

Chapter 29

Kan fibration

In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations arethe fibrations of the standard model category for simplicial sets and are therefore of fundamental importance. Kancomplexes are the fibrant objects in this model category. The name is in honor of Daniel Kan.

29.1 Definition

For each n ≥ 0, recall that the standard n -simplex, ∆n , is the representable simplicial set

∆n(i) = Hom ([i], [n])

Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological stan-dard n -simplex: the convex subspace of ℝn+1 consisting of all points (t0, . . . , tn) such that the coordinates arenon-negative and sum to 1.For each k ≤ n, this has a subcomplex Λn

k , the k-th horn inside ∆n , corresponding to the boundary of the n-simplex,with the k-th face removed. This may be formally defined in various ways, as for instance the union of the images ofthe n maps ∆n−1 → ∆n corresponding to all the other faces of ∆n .[1] Horns of the form Λ2

k sitting inside ∆2 looklike the black V at the top of the image to the right. If X is a simplicial set, then maps

s : Λnk → X

correspond to collections of n+ 1 n -simplices satisfying a compatibility condition. Explicitly, this condition can bewritten as follows. Write the n -simplices as a list (s0, . . . , sk−1, sk+1, . . . , sn+1) and require that

disj = dj−1si for all i < j with i, j = k .[2]

These conditions are satisfied for the (n− 1) -simplices of Λnk sitting inside ∆n .

A map of simplicial sets f : X → Y is a Kan fibration if, for any n ≥ 1 and 0 ≤ k ≤ n , and for any mapss : Λn

k → X and y : ∆n → Y such that f s = y i , there exists a map x : ∆n → X such that s = x i andy = f x . Stated this way, the definition is very similar to that of fibrations in topology (see also homotopy liftingproperty), whence the name “fibration”.Using the correspondence between n -simplices of a simplicial set X and morphisms ∆n → X (a consequence ofthe Yoneda lemma), this definition can be written in terms of simplices. The image of the map fs : Λn

k → Y canbe thought of as a horn as described above. Asking that fs factors through yi corresponds to requiring that there isan n -simplex in Y whose faces make up the horn from fs (together with one other face). Then the required mapx : ∆n → X corresponds to a simplex in X whose faces include the horn from s . The diagram to the right is anexample in two dimensions. Since the black V in the lower diagram is filled in by the blue 2 -simplex, if the black Vabove maps down to it then the striped blue 2 -simplex has to exist, along with the dotted blue 1 -simplex, mappingdown in the obvious way.[3]

99

100 CHAPTER 29. KAN FIBRATION

The striped blue simplex in the domain has to exist in order for this map to be a Kan fibration

29.2. EXAMPLES 101

Lifting diagram for a Kan fibration

A simplicial set X is called a Kan complex if the map from X to 1, the one-point simplicial set, is a Kan fibration. Inthe model category for simplicial sets, 1 is the terminal object and so a Kan complex is exactly the same as a fibrantobject.

29.2 Examples

An important example comes from the singular simplices used to define singular homology. Given a space X , definea singular n -simplex of X to be a continuous map from the standard topological n -simplex (as described above) toX ,

f : ∆n → X

Taking the set of these maps for all non-negative n gives a graded set,

S(X) =⨿n

Sn(X)

To make this into a simplicial set, define face maps di : Sn(X)→ Sn−1(X) by

102 CHAPTER 29. KAN FIBRATION

(dif)(t0, . . . , tn−1) = f(t0, . . . , ti−1, 0, ti, . . . , tn−1)

and degeneracy maps si : Sn(X)→ Sn+1(X) by

(sif)(t0, . . . , tn+1) = f(t0, . . . , ti−1, ti + ti+1, ti+2, . . . , tn+1)

Since the union of any n+1 faces of ∆n+1 is a strong deformation retract of ∆n+1 , any continuous function definedon these faces can be extended to ∆n+1 , which shows that S(X) is a Kan complex.[4]

It can be shown that the simplicial set underlying a simplicial group is always fibrant.

29.3 Applications

The homotopy groups of a fibrant simplicial set may be defined combinatorially, using horns, in a way that agreeswith the homotopy groups of the topological space which realizes it.

29.4 See also• Weak Kan complex (also called quasi-category, ∞-category)

• ∞-groupoid

29.5 References[1] See Goerss and Jardine, page 7

[2] See May, page 2

[3] May uses this simplicial definition; see page 25

[4] See May, page 3

29.6 Bibliography• Goerss, Paul; Jardine, John (1999). Simplicial homotopy theory. Birkhäuser. ISBN 3-7643-6064-X.

• May, Peter (1967). Simplicial objects in algebraic topology. The university of Chicago press. ISBN 0-226-51180-4.

Chapter 30

Kirkman’s schoolgirl problem

Original publication of the problem

Kirkman’s schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in1850 as Query VI in The Lady’s and Gentleman’s Diary (pg.48). The problem states:

Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required toarrange them daily so that no two shall walk twice abreast.[1]

30.1 Solution

If the girls are numbered from 01 to 15, the following arrangement is one solution:[2]

103

104 CHAPTER 30. KIRKMAN’S SCHOOLGIRL PROBLEM

A solution to this problem is an example of a Kirkman triple system,[3] which is a Steiner triple system having aparallelism, that is, a partition of the blocks of the triple system into parallel classes which are themselves partitionsof the points into disjoint blocks.There are seven non-isomorphic solutions to the schoolgirl problem.[4] Two of these are packings of the finiteprojective space PG(3,2).[5] A packing of a projective space is a partition of the lines of the space into spreads,and a spread is a partition of the points of the space into lines. These “packing” solutions can be visualized asrelations between a tetrahedron and its vertices, edges, and faces.[6]

A square, rather than tetrahedral, model may also be used:For the origin of the square model, see the Cullinane diamond theorem.

30.2 History

The first solution was published by Arthur Cayley.[7] This was shortly followed by Kirkman’s own solution[8] whichwas given as a special case of his considerations on combinatorial arrangements published three years prior.[9] J. J.Sylvester also investigated the problem and ended up declaring that Kirkman stole the idea from him. The puzzleappeared in several recreational mathematics books at the turn of the century by Lucas,[10] Rouse Ball,[11] Ahrens,[12]

and Dudeney.[13]

Kirkman often complained about the fact that his substantial paper (Kirkman 1847) was totally eclipsed by the popularinterest in the schoolgirl problem.[14]

30.3 Generalization

The problem can be generalized to n girls, where n must be an odd multiple of 3 (that is n ≡ 3 (mod 6) ), walking intriplets for 1

2 (n− 1) days, with the requirement, again, that no pair of girls walk in the same row twice. The solutionto this generalisation is a Steiner triple system, an S(2, 3, 6t + 3) with parallelism (that is, one in which each of the6t + 3 elements occurs exactly once in each block of 3-element sets), known as a Kirkman triple system.[2] It is thisgeneralization of the problem that Kirkman discussed first, while the famous special case n = 15 was only proposedlater.[9] A complete solution to the general case was published by D. K. Ray-Chaudhuri and R. M. Wilson in 1968,[15]

though it had already been solved by Lu Jiaxi ( ) in 1965,[16] but had not been published at that time.[17]

Many variations of the basic problem can be considered. Alan Hartman solves a problem of this type with therequirement that no trio walks in a row of four more than once[18] using Steiner quadruple systems.More recently a similar problem known as the Social Golfer Problem has gained interest that deals with 20 golferswho want to get to play with different people each day in groups of 4.As this is a regrouping strategy where all groups are orthogonal, this process within the problem of organising alarge group into a small groups where no two people share the same group twice can be referred to as orthogonalregrouping. However, this term is currently not commonly used and evidence suggests that there isn't a commonname for the process.

30.4 Other applications• Progressive dinner party designs

• Speed Networking events

• Cooperative learning strategy for increasing interaction within classroom teaching

• Sports Competitions

30.5 Notes[1] (Graham, Grötschel & Lovász 1995)

30.6. REFERENCES 105

[2] (Ball & Coxeter 1974)

[3] Weisstein, Eric W., “Kirkman’s Schoolgirl Problem”, MathWorld.

[4] (Cole 1922)

[5] (Hirschfeld 1985, pg.75)

[6] Falcone & Pavone 2011

[7] Cayley 1850

[8] Kirkman 1850

[9] Kirkman 1847

[10] Lucas 1883

[11] Rouse Ball 1892

[12] Ahrens 1901

[13] Dudeney 1917

[14] Cummings 1918

[15] Ray-Chaudhuri & Wilson 1971

[16] Jiaxi 1990

[17] Colbourn & Dinitz 2007, p. 13

[18] (Hartman 1980)

30.6 References• Ahrens, W. (1901), Mathematische Unterhaltungen und Spiele, Leipzig: Teubner

• Ball, W.W. Rouse; H.S.M. Coxeter (1974), Mathematical Recreations & Essays, Toronto and Buffalo: Uni-versity of Toronto Press, ISBN 0-8020-1844-0

• Cayley, A. (1850), “On the triadic arrangements of seven and fifteen things”, Phil. Mag. 37: 50–53, doi:10.1080/14786445008646550

• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, ISBN 1-58488-506-8

• Cole, F.W. (1922), “Kirkman parades”, Bulletin of the AmericanMathematical Society 28: 435–437, doi:10.1090/S0002-9904-1922-03599-9

• Cummings, L.D. (1918), “An undervalued Kirkman paper”, Bulletin of the American Mathematical Society 24:336–339, doi:10.1090/S0002-9904-1918-03086-3

• Dudeney, H.E. (1917), Amusements in Mathematics, New York: Dover

• Falcone, Giovanni; Pavone, Marco (2011), “Kirkman’s Tetrahedron and the Fifteen Schoolgirl Problem”,American Mathematical Monthly 118: 887–900, doi:10.4169/amer.math.monthly.118.10.887

• Graham, Ronald L.; Martin Grötschel, László Lovász (1995),Handbook of Combinatorics, Volume 2, Cambridge,MA: The MIT Press, ISBN 0-262-07171-1

• Hartman, Alan (1980), “Kirkman’s trombone player problem”, Ars Combinatoria 10: 19–26

• Hirschfeld, J.W.P. (1985), Finite Projective Spaces of Three Dimensions, Oxford: Oxford University Press,ISBN 0-19-853536-8

• Jiaxi, Lu (1990), Collected Works of Lu Jiaxi on Combinatorial Designs, Huhhot: Inner Mongolia People’sPress

106 CHAPTER 30. KIRKMAN’S SCHOOLGIRL PROBLEM

• Kirkman, Thomas P. (1847), “On a Problem in Combinations”, The Cambridge and Dublin MathematicalJournal (Macmillan, Barclay, and Macmillan) II: 191–204

• Kirkman, Thomas P. (1850), “Note on an unanswered prize question”, The Cambridge and Dublin Mathemat-ical Journal (Macmillan, Barclay and Macmillan) 5: 255–262

• Lucas, É. (1883), Récréations Mathématiques 2, Paris: Gauthier-Villars

• Ray-Chaudhuri, D.K.; Wilson, R.M. (1971), “Solution of Kirkman’s schoolgirl problem, in Combinatorics,University of California, Los Angeles, 1968", Proc. Sympos. Pure Math. (Providence, R.I.: American Mathe-matical Society) XIX: 187–203

• Rouse Ball, W.W. (1892), Mathematical Recreations and Essays, London: Macmillan

30.6. REFERENCES 107

The square model of PG(3,2)

Chapter 31

Kruskal–Katona theorem

In algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors ofabstract simplicial complexes. It includes as a special case the Erdős–Ko–Rado theorem and can be restated in termsof uniform hypergraphs. The theorem is named after Joseph Kruskal and Gyula O. H. Katona. It was independentlyproved by Marcel-Paul Schützenberger, but his contribution escaped notice for several years.

31.1 Statement

Given two positive integers N and i, there is a unique way to expand N as a sum of binomial coefficients as follows:

N =

(ni

i

)+

(ni−1

i− 1

)+ . . .+

(nj

j

), ni > ni−1 > . . . > nj ≥ j ≥ 1.

This expansion can be constructed by applying the greedy algorithm: set ni to be the maximal n such that N ≥(ni

),

replace N with the difference, i with i − 1, and repeat until the difference becomes zero. Define

N (i) =

(ni

i+ 1

)+

(ni−1

i

)+ . . .+

(nj

j + 1

).

31.1.1 Statement for simplicial complexes

An integral vector (f0, f1, ..., fd−1) is the f-vector of some (d− 1) -dimensional simplicial complex if and only if

0 ≤ fi ≤ f(i)i−1, 1 ≤ i ≤ d− 1.

31.1.2 Statement for uniform hypergraphs

Let A be a set consisting of N distinct i-element subsets of a fixed set U (“the universe”) and B be the set of all (i−r)-element subsets of the sets in A. Expand N as above. Then the cardinality of B is bounded below as follows:

|B| ≥(

ni

i− r

)+

(ni−1

i− r − 1

)+ . . .+

(nj

j − r

).

31.2 Ingredients of the proof

For every positive i, list all i-element subsets a1 < a2 < … ai of the setN of natural numbers in the reverse lexicographicorder. For example, for i = 3, the list begins

108

31.3. SEE ALSO 109

123, 124, 134, 234, 125, 135, 235, 145, 245, 345, . . . .

Given a vector f = (f0, f1, ..., fd−1) with positive integer components, let Δf be the subset of the power set 2Nconsisting of the empty set together with the first fi−1 i-element subsets of N in the list for i = 1, …, d. Then thefollowing conditions are equivalent:

1. Vector f is the f-vector of a simplicial complex Δ.

2. Δf is a simplicial complex.

3. fi ≤ f(i)i−1, 1 ≤ i ≤ d− 1.

The difficult implication is 1 ⇒ 2.

31.3 See also• Sperner’s theorem

31.4 References• Kruskal, J. B. (1963), “The number of simplices in a complex”, in Bellman, R., Mathematical OptimizationTechniques, University of California Press.

• Katona, G. O. H. (1968), “A theorem of finite sets”, in Erdős, P.; Katona, G. O. H., Theory of Graphs,Akadémiai Kiadó and Academic Press.

• Knuth, D., The Art of Computer Programming, pre-fascicle 3a: Generating all combinations. Contains a proofvia a more general theorem in discrete geometry.

• Stanley, Richard (1996), Combinatorics and commutative algebra, Progress in Mathematics 41 (2nd ed.),Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3836-9.

31.5 External links• Kruskal-Katona theorem on the polymath1 wiki

Chapter 32

Levi graph

In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidencestructure.[1][2] From a collection of points and lines in an incidence geometry or a projective configuration, we forma graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line.They are named for F. W. Levi, who wrote about them in 1942.[1][3]

The Levi graph of a system of points and lines usually has girth at least six: Any 4-cycles would correspond to twolines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levigraph of an abstract incidence structure.[1] Levi graphs of configurations are biregular, and every biregular graph withgirth at least six can be viewed as the Levi graph of an abstract configuration.[4]

Levi graphs may also be defined for other types of incidence structure, such as the incidences between points andplanes in Euclidean space. For every Levi graph, there is an equivalent hypergraph, and vice versa.

32.1 Examples

• The Desargues graph is the Levi graph of the Desargues configuration, composed of 10 points and 10 lines.There are 3 points on each line, and 3 lines passing through each point. The Desargues graph can also be viewedas the generalized Petersen graph G(10,3) or the bipartite Kneser graph with parameters 5,2. It is 3-regularwith 20 vertices.

• The Heawood graph is the Levi graph of the Fano plane. It is also known as the (3,6)-cage, and is 3-regularwith 14 vertices.

• The Möbius–Kantor graph is the Levi graph of the Möbius–Kantor configuration, a system of 8 points and 8lines that cannot be realized by straight lines in the Euclidean plane. It is 3-regular with 16 vertices.

• The Pappus graph is the Levi graph of the Pappus configuration, composed of 9 points and 9 lines. Like theDesargues configuration there are 3 points on each line and 3 lines passing through each point. It is 3-regularwith 18 vertices.

• The Gray graph is the Levi graph of a configuration that can be realized in R3 as a 3×3×3 grid of 27 pointsand the 27 orthogonal lines through them.

• The Tutte eight-cage is the Levi graph of the Cremona–Richmond configuration. It is also known as the (3,8)-cage, and is 3-regular with 30 vertices.

• The four-dimensional hypercube graph Q4 is the Levi graph of the Möbius configuration formed by the pointsand planes of two mutually incident tetrahedra.

• The Ljubljana graph on 112 vertices is the Levi graph of the Ljubljana configuration.[5]

110

32.2. REFERENCES 111

32.2 References[1] Grünbaum, Branko (2006), “Configurations of points and lines”, The Coxeter Legacy, Providence, RI: American Mathe-

matical Society, pp. 179–225, MR 2209028. See in particular p. 181.

[2] Polster, Burkard (1998), A Geometrical Picture Book, Universitext, New York: Springer-Verlag, p. 5, doi:10.1007/978-1-4419-8526-2, ISBN 0-387-98437-2, MR 1640615.

[3] Levi, F. W. (1942), Finite Geometrical Systems, Calcutta: University of Calcutta, MR 0006834.

[4] Gropp, Harald (2007), “VI.7 Configurations”, in Colbourn, Charles J.; Dinitz, Jeffrey H., Handbook of combinatorialdesigns, Discrete Mathematics and its Applications (Boca Raton) (Second ed.), Chapman & Hall/CRC, Boca Raton, FL,pp. 353–355.

[5] Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; Potočnik, P. (2002), The Ljubljana Graph (PDF), IMFM Preprint40-845, University of Ljubljana Department of Mathematics.

32.3 External links• Weisstein, Eric W., “Levi Graph”, MathWorld.

Chapter 33

Matroid

In combinatorics, a branch of mathematics, amatroid /ˈmeɪtrɔɪd/ is a structure that captures and generalizes the notionof linear independence in vector spaces. There are many equivalent ways to define a matroid, the most significantbeing in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is theabstraction of various notions of central importance in these fields. Matroids have found applications in geometry,topology, combinatorial optimization, network theory and coding theory.[1][2]

33.1 Definition

There are many equivalent (cryptomorphic) ways to define a (finite) matroid.[3]

33.1.1 Independent sets

In terms of independence, a finite matroid M is a pair (E, I) , where E is a finite set (called the ground set) and Iis a family of subsets of E (called the independent sets) with the following properties:[4]

1. The empty set is independent, i.e., ∅ ∈ I . Alternatively, at least one subset of E is independent, i.e., I = ∅ .

2. Every subset of an independent set is independent, i.e., for each A′ ⊂ A ⊂ E , if A ∈ I then A′ ∈ I . Thisis sometimes called the hereditary property.

3. If A and B are two independent sets of I and A has more elements than B , then there exists an element inA that when added to B gives a larger independent set than B . This is sometimes called the augmentationproperty or the independent set exchange property.

The first two properties define a combinatorial structure known as an independence system.

33.1.2 Bases and circuits

A subset of the ground set E that is not independent is called dependent. A maximal independent set—that is,an independent set which becomes dependent on adding any element of E —is called a basis for the matroid. Acircuit in a matroid M is a minimal dependent subset of E —that is, a dependent set whose proper subsets areall independent. The terminology arises because the circuits of graphic matroids are cycles in the correspondinggraphs.[4]

The dependent sets, the bases, or the circuits of a matroid characterize the matroid completely: a set is independentif and only if it is not dependent, if and only if it is a subset of a basis, and if and only if it does not contain a circuit.The collection of dependent sets, or of bases, or of circuits each has simple properties that may be taken as axiomsfor a matroid. For instance, one may define a matroid M to be a pair (E,B) , where E is a finite set as before and Bis a collection of subsets of E , called “bases”, with the following properties:[4]

112

33.1. DEFINITION 113

1. B is nonempty.

2. If A and B are distinct members of B and a ∈ A \ B , then there exists an element b ∈ B \ A such thatA \ a ∪ b ∈ B . (Here the backslash symbol stands for the difference of sets. This property is called thebasis exchange property.)

It follows from the basis exchange property that no member of B can be a proper subset of another.

33.1.3 Rank functions

It is a basic result of matroid theory, directly analogous to a similar theorem of bases in linear algebra, that any twobases of a matroid M have the same number of elements. This number is called the rank of M . If M is a matroidon E , and A is a subset of E , then a matroid on A can be defined by considering a subset of A to be independent ifand only if it is independent in M . This allows us to talk about submatroids and about the rank of any subset of E. The rank of a subset A is given by the rank function r(A) of the matroid, which has the following properties:[4]

• The value of the rank function is always a non-negative integer.

• For any subset A of E , r(A) ≤ |A| .

• For any two subsets A and B of E , r(A∪B)+ r(A∩B) ≤ r(A)+ r(B) . That is, the rank is a submodularfunction.

• For any set A and element x , r(A) ≤ r(A ∪ x) ≤ r(A) + 1 . From the first of these two inequalities itfollows more generally that, if A ⊂ B ⊂ E , then r(A) ≤ r(B) ≤ r(E) . That is, the rank is a monotonicfunction.

These properties can be used as one of the alternative definitions of a finite matroid: if (E, r) satisfies these properties,then the independent sets of a matroid over E can be defined as those subsets A of E with r(A) = |A| .The difference |A| − r(A) is called the nullity or corank of the subset A . It is the minimum number of elementsthat must be removed from A to obtain an independent set. The nullity of E in M is called the nullity or corank ofM .

33.1.4 Closure operators

Let M be a matroid on a finite set E , with rank function r as above. The closure cl(A) of a subset A of E is the set

cl(A) =x ∈ E | r(A) = r

(A ∪ x

)This defines a closure operator cl : P(E)→ P(E) where P denotes the power set, with the following properties:

• For all subsets X of E , X ⊆ cl(X) .

• For all subsets X of E , cl(X) = cl(cl(X)) .

• For all subsets X and Y of E with X ⊆ Y , cl(X) ⊆ cl(Y ) .

• For all elements a , and b of E and all subsets Y of E , if a ∈ cl(Y ∪b)\cl(Y ) then b ∈ cl(Y ∪a)\cl(Y ).

The first three of these properties are the defining properties of a closure operator. The fourth is sometimes calledthe Mac Lane–Steinitz exchange property. These properties may be taken as another definition of matroid: everyfunction cl : P(E)→ P(E) that obeys these properties determines a matroid.[4]

114 CHAPTER 33. MATROID

33.1.5 Flats

A set whose closure equals itself is said to be closed, or a flat or subspace of the matroid.[5] A set is closed if it ismaximal for its rank, meaning that the addition of any other element to the set would increase the rank. The closedsets of a matroid are characterized by a covering partition property:

• The whole point set E is closed.

• If S and T are flats, then S ∩ T is a flat.

• If S is a flat, then the flats T that cover S (meaning that T properly contains S but there is no flat U betweenS and T ), partition the elements of E \ S .

The class L(M) of all flats, partially ordered by set inclusion, forms a matroid lattice. Conversely, every matroidlattice L forms a matroid over its set E of atoms under the following closure operator: for a set S of atoms with join∨S ,

cl(S) = x ∈ E | x ≤∨

S

The flats of this matroid correspond one-for-one with the elements of the lattice; the flat corresponding to latticeelement y is the set

x ∈ E | x ≤ y

Thus, the lattice of flats of this matroid is naturally isomorphic to L .

33.1.6 Hyperplanes

In a matroid of rank r , a flat of rank r − 1 is called a hyperplane. These are the maximal proper flats; that is, theonly superset of a hyperplane that is also a flat is the set E of all the elements of the matroid. Hyperplanes are alsocalled coatoms or copoints. An equivalent definition: A coatom is a subset of E that does not span M, but such thatadding any other element to it does make a spanning set.[6]

The family H of hyperplanes of a matroid has the following properties, which may be taken as yet another axioma-tization of matroids:[6]

• There do not exist distinct sets X and Y inH with X ⊂ Y . That is, the hyperplanes form a Sperner family.

• For every x ∈ E and Y,Z ∈ H with x /∈ Y ∪ Z , there exists X ∈ H with (Y ∩ Z) ∪ x ⊆ X .

33.2 Examples

33.2.1 Uniform matroids

Let E be a finite set and k a natural number. One may define a matroid on E by taking every k-element subset of Eto be a basis. This is known as the uniform matroid of rank k. A uniform matroid with rank k and with n elementsis denoted Uk,n . All uniform matroids of rank at least 2 are simple. The uniform matroid of rank 2 on n points iscalled the n-point line. A matroid is uniform if and only if it has no circuits of size less than the one plus the rankof the matroid. The direct sums of uniform matroids are called partition matroids.In the uniform matroid U0,n , every element is a loop (an element that does not belong to any independent set), andin the uniform matroid Un,n , every element is a coloop (an element that belongs to all bases). The direct sum ofmatroids of these two types is a partition matroid in which every element is a loop or a coloop; it is called a discretematroid. An equivalent definition of a discrete matroid is a matroid in which every proper, non-empty subset of theground set E is a separator.

33.2. EXAMPLES 115

The Fano matroid, derived from the Fano plane. It is GF(2)-linear but not real-linear.

33.2.2 Matroids from linear algebra

Matroid theory developed mainly out of a deep examination of the properties of independence and dimension invector spaces. There are two ways to present the matroids defined in this way:

• If E is any finite subset of a vector space V, then we can define a matroidM on E by taking the independent setsof M to be the linearly independent subsets of E. The validity of the independent set axioms for this matroidfollows from the Steinitz exchange lemma. If M is a matroid that can be defined in this way, we say the set Erepresents M. Matroids of this kind are called vector matroids. An important example of a matroid definedin this way is the Fano matroid, a rank-three matroid derived from the Fano plane, a finite geometry with sevenpoints (the seven elements of the matroid) and seven lines (the nontrivial flats of the matroid). It is a linearmatroid whose elements may be described as the seven nonzero points in a three-dimensional vector space overthe finite field GF(2). However, it is not possible to provide a similar representation for the Fano matroid usingthe real numbers in place of GF(2).

• A matrixAwith entries in a field gives rise to a matroidM on its set of columns. The dependent sets of columnsin the matroid are those that are linearly dependent as vectors. This matroid is called the column matroid ofA, and A is said to representM. For instance, the Fano matroid can be represented in this way as a 3 × 7 (0,1)-matrix. Column matroids are just vector matroids under another name, but there are often reasons to favor the

116 CHAPTER 33. MATROID

The Vámos matroid, not linear over any field

matrix representation. (There is one technical difference: a column matroid can have distinct elements that arethe same vector, but a vector matroid as defined above cannot. Usually this difference is insignificant and canbe ignored, but by letting E be a multiset of vectors one brings the two definitions into complete agreement.)

33.2. EXAMPLES 117

A matroid that is equivalent to a vector matroid, although it may be presented differently, is called representable orlinear. If M is equivalent to a vector matroid over a field F, then we say M is representable over F ; in particular,M is real-representable if it is representable over the real numbers. For instance, although a graphic matroid (seebelow) is presented in terms of a graph, it is also representable by vectors over any field. A basic problem in matroidtheory is to characterize the matroids that may be represented over a given field F; Rota’s conjecture describes apossible characterization for every finite field. The main results so far are characterizations of binary matroids (thoserepresentable over GF(2)) due to Tutte (1950s), of ternary matroids (representable over the 3-element field) due toReid and Bixby, and separately to Seymour (1970s), and of quaternary matroids (representable over the 4-elementfield) due to Geelen, Gerards, and Kapoor (2000). This is very much an open area.A regular matroid is a matroid that is representable over all possible fields. The Vámos matroid is the simplest exampleof a matroid that is not representable over any field.

33.2.3 Matroids from graph theory

A second original source for the theory of matroids is graph theory.Every finite graph (or multigraph) G gives rise to a matroid M(G) as follows: take as E the set of all edges in G andconsider a set of edges independent if and only if it is a forest; that is, if it does not contain a simple cycle. ThenM(G) is called a cycle matroid. Matroids derived in this way are graphic matroids. Not every matroid is graphic,but all matroids on three elements are graphic.[7] Every graphic matroid is regular.Other matroids on graphs were discovered subsequently:

• The bicircular matroid of a graph is defined by calling a set of edges independent if every connected subsetcontains at most one cycle.

• In any directed or undirected graph G let E and F be two distinguished sets of vertices. In the set E, define asubset U to be independent if there are |U | vertex-disjoint paths from F onto U. This defines a matroid on Ecalled a gammoid:[8] a strict gammoid is one for which the set E is the whole vertex set of G.[9]

• In a bipartite graph G = (U,V,E), one may form a matroid in which the elements are vertices on one side Uof the bipartition, and the independent subsets are sets of endpoints of matchings of the graph. This is calleda transversal matroid,[10][11] and it is a special case of a gammoid.[8] The transversal matroids are the dualmatroids to the strict gammoids.[9]

• Graphic matroids have been generalized to matroids from signed graphs, gain graphs, and biased graphs. AgraphG with a distinguished linear classB of cycles, known as a “biased graph” (G,B), has two matroids, knownas the frame matroid and the lift matroid of the biased graph. If every cycle belongs to the distinguishedclass, these matroids coincide with the cycle matroid of G. If no cycle is distinguished, the frame matroid is thebicircular matroid of G. A signed graph, whose edges are labeled by signs, and a gain graph, which is a graphwhose edges are labeled orientably from a group, each give rise to a biased graph and therefore have frame andlift matroids.

• The Laman graphs form the bases of the two-dimensional rigidity matroid, a matroid defined in the theory ofstructural rigidity.

• Let G be a connected graph and E be its edge set. Let I be the collection of subsets F of E such that G − F isstill connected. Then M∗(G) = (E, I) is a matroid, called bond matroid of G. Note that the rank functionr(F) is the number of minimal cycles in the subgraph induced on the edge subset F.

33.2.4 Matroids from field extensions

A third original source of matroid theory is field theory.An extension of a field gives rise to a matroid. Suppose F and K are fields with K containing F. Let E be any finitesubset of K. Define a subset S of E to be algebraically independent if the extension field F(S) has transcendence degreeequal to |S|.[12]

118 CHAPTER 33. MATROID

A matroid that is equivalent to a matroid of this kind is called an algebraicmatroid.[13] The problem of characterizingalgebraic matroids is extremely difficult; little is known about it. The Vámos matroid provides an example of a matroidthat is not algebraic.

33.3 Basic constructions

There are some standard ways to make new matroids out of old ones.

33.3.1 Duality

If M is a finite matroid, we can define the orthogonal or dual matroid M* by taking the same underlying set andcalling a set a basis in M* if and only if its complement is a basis in M. It is not difficult to verify that M* is a matroidand that the dual of M* is M.[14]

The dual can be described equally well in terms of other ways to define a matroid. For instance:

• A set is independent in M* if and only if its complement spans M.

• A set is a circuit of M* if and only if its complement is a coatom in M.

• The rank function of the dual is r∗(S) = |S| − r(M) + r (E \ S) .

According to a matroid version of Kuratowski’s theorem, the dual of a graphic matroid M is a graphic matroid if andonly if M is the matroid of a planar graph. In this case, the dual of M is the matroid of the dual graph of G.[15] Thedual of a vector matroid representable over a particular field F is also representable over F. The dual of a transversalmatroid is a strict gammoid and vice versa.ExampleThe cycle matroid of a graph is the dual matroid of its bond matroid.

33.3.2 Minors

If M is a matroid with element set E, and S is a subset of E, the restriction of M to S, written M |S, is the matroid onthe set S whose independent sets are the independent sets ofM that are contained in S. Its circuits are the circuits ofMthat are contained in S and its rank function is that of M restricted to subsets of S. In linear algebra, this correspondsto restricting to the subspace generated by the vectors in S. Equivalently if T = M−S this may be termed the deletionof T, written M\T or M−T. The submatroids of M are precisely the results of a sequence of deletions: the order isirrelevant.[16][17]

The dual operation of restriction is contraction.[18] If T is a subset of E, the contraction of M by T, written M/T, isthe matroid on the underlying set E − T whose rank function is r′(A) = r(A ∪ T ) − r(T ). [19] In linear algebra,this corresponds to looking at the quotient space by the linear space generated by the vectors in T, together with theimages of the vectors in E - T.A matroid N that is obtained from M by a sequence of restriction and contraction operations is called a minor ofM.[17][20] We say M contains N as a minor. Many important families of matroids may be characterized by theminor-minimal matroids that do not belong to the family; these are called forbidden or excluded minors.[21]

33.3.3 Sums and unions

Let M be a matroid with an underlying set of elements E, and let N be another matroid on an underlying set F. Thedirect sum of matroids M and N is the matroid whose underlying set is the disjoint union of E and F, and whoseindependent sets are the disjoint unions of an independent set of M with an independent set of N.The union ofM andN is the matroid whose underlying set is the union (not the disjoint union) of E and F, and whoseindependent sets are those subsets which are the union of an independent set in M and one in N. Usually the term“union” is applied when E = F, but that assumption is not essential. If E and F are disjoint, the union is the directsum.

33.4. ADDITIONAL TERMINOLOGY 119

33.4 Additional terminology

Let M be a matroid with an underlying set of elements E.

• E may be called the ground set of M. Its elements may be called the points of M.

• A subset of E spans M if its closure is E. A set is said to span a closed set K if its closure is K.

• The girth of a matroid is the size of its smallest circuit or dependent set.

• An element that forms a single-element circuit of M is called a loop. Equivalently, an element is a loop if itbelongs to no basis.[7][22]

• An element that belongs to no circuit is called a coloop or isthmus. Equivalently, an element is a coloop if itbelongs to every basis. Loop and coloops are mutually dual.[22]

• If a two-element set f, g is a circuit of M, then f and g are parallel in M.[7]

• A matroid is called simple if it has no circuits consisting of 1 or 2 elements. That is, it has no loops and noparallel elements. The term combinatorial geometry is also used.[7] A simple matroid obtained from anothermatroidM by deleting all loops and deleting one element from each 2-element circuit until no 2-element circuitsremain is called a simplification of M.[23] A matroid is co-simple if its dual matroid is simple.[24]

• A union of circuits is sometimes called a cycle of M. A cycle is therefore the complement of a flat of the dualmatroid. (This usage conflicts with the common meaning of “cycle” in graph theory.)

• A separator of M is a subset S of E such that r(S)+r(E−S) = r(M) . A proper or non-trivial separatoris a separator that is neither E nor the empty set.[25] An irreducible separator is a separator that contains noother non-empty separator. The irreducible separators partition the ground set E.

• A matroid which cannot be written as the direct sum of two nonempty matroids, or equivalently which hasno proper separators, is called connected or irreducible. A matroid is connected if and only if its dual isconnected.[26]

• A maximal irreducible submatroid of M is called a component of M. A component is the restriction of M toan irreducible separator, and contrariwise, the restriction of M to an irreducible separator is a component. Aseparator is a union of components.[25]

• A matroid M is called a frame matroid if it, or a matroid that contains it, has a basis such that all the pointsof M are contained in the lines that join pairs of basis elements.[27]

• A matroid is called a paving matroid if all of its circuits have size at least equal to its rank.[28]

• The matroid polytope PM is the convex hull of the indicator vectors of the bases of M .

33.5 Algorithms

33.5.1 Greedy algorithm

A weighted matroid is a matroid together with a function from its elements to the nonnegative real numbers. Theweight of a subset of elements is defined to be the sum of the weights of the elements in the subset. The greedyalgorithm can be used to find a maximum-weight basis of the matroid, by starting from the empty set and repeatedlyadding one element at a time, at each step choosing a maximum-weight element among the elements whose additionwould preserve the independence of the augmented set.[29] This algorithm does not need to know anything about thedetails of the matroid’s definition, as long as it has access to the matroid through an independence oracle, a subroutinefor testing whether a set is independent.This optimization algorithm may be used to characterize matroids: if a family F of sets, closed under taking subsets,has the property that, no matter how the sets are weighted, the greedy algorithm finds a maximum-weight set in thefamily, then F must be the family of independent sets of a matroid.[30]

The notion of matroid has been generalized to allow for other types of sets on which a greedy algorithm give optimalsolutions; see greedoid and matroid embedding for more information.

120 CHAPTER 33. MATROID

33.5.2 Matroid partitioning

The matroid partitioning problem is to partition the elements of a matroid into as few independent sets as possible, andthe matroid packing problem is to find as many disjoint spanning sets as possible. Both can be solved in polynomialtime, and can be generalized to the problem of computing the rank or finding an independent set in a matroid sum.

33.5.3 Matroid intersection

The intersection of two or more matroids is the family of sets that are simultaneously independent in each of thematroids. The problem of finding the largest set, or the maximum weighted set, in the intersection of two matroids canbe found in polynomial time,and provides a solution to many other important combinatorial optimization problems.For instance, maximum matching in bipartite graphs can be expressed as a problem of intersecting two partitionmatroids. However, finding the largest set in an intersection of three or more matroids is NP-complete.

33.5.4 Matroid software

Two standalone systems for calculations with matroids are Kingan’s Oid and Hlineny’s Macek. Both of them are opensourced packages. “Oid” is an interactive, extensible software system for experimenting with matroids. “Macek”is a specialized software system with tools and routines for reasonably efficient combinatorial computations withrepresentable matroids.SAGE, the open source mathematics software system, contains a matroid package.

33.6 Polynomial invariants

There are two especially significant polynomials associated to a finite matroid M on the ground set E. Each is amatroid invariant, which means that isomorphic matroids have the same polynomial.

33.6.1 Characteristic polynomial

The characteristic polynomial of M (which is sometimes called the chromatic polynomial,[31] although it does notcount colorings), is defined to be

pM (λ) :=∑S⊆E

(−1)|S|λr(M)−r(S),

or equivalently (as long as the empty set is closed in M) as

pM (λ) :=∑A

µ(∅, A)λr(M)−r(A) ,

where μ denotes the Möbius function of the geometric lattice of the matroid.[32]

When M is the cycle matroid M(G) of a graph G, the characteristic polynomial is a slight transformation of thechromatic polynomial, which is given by χG (λ) = λcpM ₍G₎ (λ), where c is the number of connected components ofG.When M is the bond matroid M*(G) of a graph G, the characteristic polynomial equals the flow polynomial of G.When M is the matroid M(A) of an arrangement A of linear hyperplanes in Rn (or Fn where F is any field), thecharacteristic polynomial of the arrangement is given by pA (λ) = λn−r(M)pM ₍A₎ (λ).

Beta invariant

The beta invariant of a matroid, introduced by Crapo (1967), may be expressed in terms of the characteristicpolynomial p as an evaluation of the derivative[33]

33.6. POLYNOMIAL INVARIANTS 121

β(M) = (−1)r(M)−1p′M (1)

or directly as[34]

β(M) = (−1)r(M)∑X⊆E

(−1)|X|r(X) .

The beta invariant is non-negative, and is zero if and only if M is disconnected, or empty, or a loop. Otherwise itdepends only on the lattice of flats of M. If M has no loops and coloops then β(M) = β(M∗).[34]

33.6.2 Tutte polynomial

The Tutte polynomial of a matroid, TM (x,y), generalizes the characteristic polynomial to two variables. This givesit more combinatorial interpretations, and also gives it the duality property

TM∗(x, y) = TM (y, x),

which implies a number of dualities between properties of M and properties of M *. One definition of the Tuttepolynomial is

TM (x, y) =∑S⊆E

(x− 1)r(M)−r(S)(y − 1)|S|−r(S).

This expresses the Tutte polynomial as an evaluation of the corank-nullity or rank generating polynomial,[35]

RM (u, v) =∑S⊆E

ur(M)−r(S)v|S|−r(S).

From this definition it is easy to see that the characteristic polynomial is, up to a simple factor, an evaluation of TM,specifically,

pM (λ) = (−1)r(M)TM (1− λ, 0).

Another definition is in terms of internal and external activities and a sum over bases, reflecting the fact that T(1,1)is the number of bases.[36] This, which sums over fewer subsets but has more complicated terms, was Tutte’s originaldefinition.There is a further definition in terms of recursion by deletion and contraction.[37] The deletion-contraction identity is

F (M) = F (M − e) + F (M/e) when e is neither a loop nor a coloop.

An invariant of matroids (i.e., a function that takes the same value on isomorphic matroids) satisfying this recursionand the multiplicative condition

F (M ⊕M ′) = F (M)F (M ′)

is said to be a Tutte-Grothendieck invariant.[35] The Tutte polynomial is the most general such invariant; thatis, the Tutte polynomial is a Tutte-Grothendieck invariant and every such invariant is an evaluation of the Tuttepolynomial.[31]

The Tutte polynomial TG of a graph is the Tutte polynomial TM ₍G₎ of its cycle matroid.

122 CHAPTER 33. MATROID

33.7 Infinite matroids

The theory of infinite matroids is much more complicated than that of finite matroids and forms a subject of its own.For a long time, one of the difficulties has been that there were many reasonable and useful definitions, none of whichappeared to capture all the important aspects of finite matroid theory. For instance, it seemed to be hard to havebases, circuits, and duality together in one notion of infinite matroids.The simplest definition of an infinite matroid is to require finite rank; that is, the rank of E is finite. This theory issimilar to that of finite matroids except for the failure of duality due to the fact that the dual of an infinite matroidof finite rank does not have finite rank. Finite-rank matroids include any subsets of finite-dimensional vector spacesand of field extensions of finite transcendence degree.The next simplest infinite generalization is finitary matroids. A matroid is finitary if it has the property that

x ∈ cl(Y )⇔ (∃Y ′ ⊆ Y )Y ′ and finite is x ∈ cl(Y ′).

Equivalently, every dependent set contains a finite dependent set. Examples are linear dependence of arbitrary subsetsof infinite-dimensional vector spaces (but not infinite dependencies as in Hilbert and Banach spaces), and algebraicdependence in arbitrary subsets of field extensions of possibly infinite transcendence degree. Again, the class offinitary matroid is not self-dual, because the dual of a finitary matroid is not finitary. Finitary infinite matroids arestudied in model theory, a branch of mathematical logic with strong ties to algebra.In the late 1960s matroid theorists asked for a more general notion that shares the different aspects of finite matroidsand generalizes their duality. Many notions of infinite matroids were defined in response to this challenge, but thequestion remained open. One of the approaches examined by D.A. Higgs became known as B-matroids and wasstudied by Higgs, Oxley and others in the 1960s and 1970s. According to a recent result by Bruhn, Diestel, andKriesell et al. (2013), it solves the problem: Arriving at the same notion independently, they provided five equivalentsystems of axioms – in terms of independence, bases, circuits, closure and rank. The duality of B-matroids generalizesdualities that can be observed in infinite graphs.The independence axioms are as follows:

1. The empty set is independent.

2. Every subset of an independent set is independent.

3. For every nonmaximal (under set inclusion) independent set I and maximal independent set J, there is x ∈ J \Isuch that I ∪ x is independent.

4. For every subsetX of the base space, every independent subset I ofX can be extended to a maximal independentsubset of X.

With these axioms, every matroid has a dual.

33.8 History

Matroid theory was introduced by Hassler Whitney (1935). It was also independently discovered by Takeo Nakasawa,whose work was forgotten for many years (Nishimura & Kuroda 2009).In his seminal paper, Whitney provided two axioms for independence, and defined any structure adhering to theseaxioms to be “matroids”. (Although it was perhaps implied, he did not include an axiom requiring at least one subsetto be independent.) His key observation was that these axioms provide an abstraction of “independence” that iscommon to both graphs and matrices. Because of this, many of the terms used in matroid theory resemble the termsfor their analogous concepts in linear algebra or graph theory.Almost immediately after Whitney first wrote about matroids, an important article was written by Saunders MacLane (1936) on the relation of matroids to projective geometry. A year later, B. L. van der Waerden (1937) notedsimilarities between algebraic and linear dependence in his classic textbook on Modern Algebra.In the 1940s Richard Rado developed further theory under the name “independence systems” with an eye towardstransversal theory, where his name for the subject is still sometimes used.

33.9. RESEARCHERS 123

In the 1950s W. T. Tutte became the foremost figure in matroid theory, a position he retained for many years.His contributions were plentiful, including the characterization of binary, regular, and graphic matroids by excludedminors; the regular-matroid representability theorem; the theory of chain groups and their matroids; and the tools heused to prove many of his results, the “Path theorem” and "Homotopy theorem" (see, e.g., Tutte 1965), which areso complex that later theorists have gone to great trouble to eliminate the necessity of using them in proofs. (A fineexample is A. M. H. Gerards' short proof (1989) of Tutte’s characterization of regular matroids.)Henry Crapo (1969) and Thomas Brylawski (1972) generalized to matroids Tutte’s “dichromate”, a graphic polyno-mial now known as the Tutte polynomial (named by Crapo). Their work has recently (especially in the 2000s) beenfollowed by a flood of papers—though not as many as on the Tutte polynomial of a graph.In 1976 Dominic Welsh published the first comprehensive book on matroid theory.Paul Seymour's decomposition theorem for regular matroids (1980) was the most significant and influential work ofthe late 1970s and the 1980s. Another fundamental contribution, by Kahn & Kung (1982), showed why projectivegeometries and Dowling geometries play such an important role in matroid theory.By this time there were many other important contributors, but one should not omit to mention Geoff Whittle'sextension to ternary matroids of Tutte’s characterization of binary matroids that are representable over the rationals(Whittle 1995), perhaps the biggest single contribution of the 1990s. In the current period (since around 2000) theMatroid Minors Project of Jim Geelen, Gerards, Whittle, and others, which attempts to duplicate for matroids thatare representable over a finite field the success of the Robertson–Seymour Graph Minors Project (see Robertson–Seymour theorem), has produced substantial advances in the structure theory of matroids. Many others have alsocontributed to that part of matroid theory, which (in the first and second decades of the 21st century) is flourishing.

33.9 Researchers

Mathematicians who pioneered the study of matroids include Takeo Nakasawa,[38] Saunders Mac Lane, RichardRado, W. T. Tutte, B. L. van der Waerden, and Hassler Whitney. Other major contributors include Jack Edmonds,Jim Geelen, Eugene Lawler, László Lovász, Gian-Carlo Rota, P. D. Seymour, and Dominic Welsh.There is an on-line list of current researchers.

33.10 See also• Antimatroid

• Coxeter matroid

• Oriented matroid

• Pregeometry (model theory)

• Polymatroid

• Greedoid

33.11 Notes[1] Neel, David L.; Neudauer, Nancy Ann (2009). “Matroids you have known” (PDF). Mathematics Magazine 82 (1): 26–41.

doi:10.4169/193009809x469020. Retrieved 4 October 2014.

[2] Kashyap, Navin; Soljanin, Emina; Vontobel, Pascal. “Applications of Matroid Theory and Combinatorial Optimization toInformation and Coding Theory” (PDF). www.birs.ca. Retrieved 4 October 2014.

[3] A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh(1976). See Bryzlawski’s appendix in White (1986) pp.298–302 for a list of equivalent axiom systems.

[4] Welsh (1976), Section 1.2, “Axiom Systems for a Matroid”, pp. 7–9.

[5] Welsh (1976), Section 1.8, “Closed sets = Flats = Subspaces”, pp. 21–22.

124 CHAPTER 33. MATROID

[6] Welsh (1976), Section 2.2, “The Hyperplanes of a Matroid”, pp. 38–39.

[7] Oxley 1992, p. 13

[8] Oxley 1992, pp. 115

[9] Oxley 1992, p. 100

[10] Oxley 1992, pp. 46–48

[11] 1987

[12] Oxley 1992, p. 215

[13] Oxley 1992, p. 216

[14] White 1986, p. 32

[15] White 1986, p. 105

[16] White 1986, p. 131

[17] White 1986, p. 224

[18] White 1986, p. 139

[19] White 1986, p. 140

[20] White 1986, p. 150

[21] White 1986, pp. 146–147

[22] White 1986, p. 130

[23] Oxley 1992, p. 52

[24] Oxley 1992, p. 347

[25] Oxley 1992, p. 128

[26] White 1986, p. 110

[27] Zaslavsky, Thomas (1994). “Frame matroids and biased graphs”. Eur. J. Comb. 15 (3): 303–307. doi:10.1006/eujc.1994.1034.ISSN 0195-6698. Zbl 0797.05027.

[28] Oxley 1992, p. 26

[29] Oxley 1992, p. 63

[30] Oxley 1992, p. 64

[31] White 1987, p. 127

[32] White 1987, p. 120

[33] White 1987, p. 123

[34] White 1987, p. 124

[35] White 1987, p. 126

[36] White 1992, p. 188

[37] White 1986, p. 260

[38] Nishimura & Kuroda (2009).

33.12. REFERENCES 125

33.12 References

• Bruhn, Henning; Diestel, Reinhard; Kriesell, Matthias; Pendavingh, Rudi; Wollan, Paul (2013), “Axioms forinfinite matroids”, Advances in Mathematics 239: 18–46, arXiv:1003.3919, doi:10.1016/j.aim.2013.01.011,MR 3045140.

• Bryant, Victor; Perfect, Hazel (1980), Independence Theory in Combinatorics, London and New York: Chap-man and Hall, ISBN 0-412-22430-5.

• Brylawski, Thomas H. (1972), “A decomposition for combinatorial geometries”, Transactions of the Amer-ican Mathematical Society (American Mathematical Society) 171: 235–282, doi:10.2307/1996381, JSTOR1996381.

• Crapo, Henry H. (1969), “The Tutte polynomial”,AequationesMathematicae 3 (3): 211–229, doi:10.1007/BF01817442.

• Crapo, Henry H.; Rota, Gian-Carlo (1970), On the Foundations of Combinatorial Theory: Combinatorial Ge-ometries, Cambridge, Mass.: M.I.T. Press, ISBN 978-0-262-53016-3, MR 0290980.

• Geelen, Jim; Gerards, A. M. H.; Whittle, Geoff (2007), “Towards a matroid-minor structure theory”, in Grim-mett, Geoffrey (ed.) et al., Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh, OxfordLecture Series in Mathematics and its Applications 34, Oxford: Oxford University Press, pp. 72–82.

• Gerards, A. M. H. (1989), “A short proof of Tutte’s characterization of totally unimodular matrices”, LinearAlgebra and its Applications, 114/115: 207–212, doi:10.1016/0024-3795(89)90461-8.

• Kahn, Jeff; Kung, Joseph P. S. (1982), “Varieties of combinatorial geometries”, Transactions of the AmericanMathematical Society (American Mathematical Society) 271 (2): 485–499, doi:10.2307/1998894, JSTOR1998894.

• Kingan, Robert; Kingan, Sandra (2005), “A software system for matroids”, Graphs and Discovery, DIMACSSeries in Discrete Mathematics and Theoretical Computer Science, pp. 287–296.

• Kung, Joseph P. S., ed. (1986), A Source Book in Matroid Theory, Boston: Birkhäuser, ISBN 0-8176-3173-9,MR 0890330.

• Mac Lane, Saunders (1936), “Some interpretations of abstract linear dependence in terms of projective geome-try”,American Journal ofMathematics (The Johns Hopkins University Press) 58 (1): 236–240, doi:10.2307/2371070,JSTOR 2371070.

• Nishimura, Hirokazu; Kuroda, Susumu, eds. (2009), A lost mathematician, Takeo Nakasawa. The forgottenfather of matroid theory, Basel: Birkhäuser Verlag, ISBN 978-3-7643-8572-9, MR 2516551, Zbl 1163.01001.

• Oxley, James (1992), Matroid Theory, Oxford: Oxford University Press, ISBN 0-19-853563-5, MR 1207587,Zbl 0784.05002.

• Recski, András (1989), Matroid Theory and its Applications in Electric Network Theory and in Statics, Berlinand Budapest: Springer-Verlag and Akademiai Kiado, ISBN 3-540-15285-7, MR 1027839.

• Sapozhenko, A.A. (2001), “M/m062870”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Seymour, Paul D. (1980), “Decomposition of regular matroids”, Journal of Combinatorial Theory, Series B 28(3): 305–359, doi:10.1016/0095-8956(80)90075-1, Zbl 0443.05027.

• Truemper, Klaus (1992),MatroidDecomposition, Boston: Academic Press, ISBN 0-12-701225-7, MR 1170126.

• Tutte, W. T. (1959), “Matroids and graphs”, Transactions of the American Mathematical Society (AmericanMathematical Society) 90 (3): 527–552, doi:10.2307/1993185, JSTOR 1993185, MR 0101527.

• Tutte, W. T. (1965), “Lectures on matroids”, Journal of Research of the National Bureau of Standards (U.S.A.),Sect. B 69: 1–47.

• Tutte, W.T. (1971), Introduction to the theory of matroids, Modern Analytic and Computational Methods inScience and Mathematics 37, New York: American Elsevier Publishing Company, Zbl 0231.05027.

126 CHAPTER 33. MATROID

• Vámos, Peter (1978), “The missing axiom of matroid theory is lost forever”, Journal of the London Mathemat-ical Society 18 (3): 403–408, doi:10.1112/jlms/s2-18.3.403.

• van der Waerden, B. L. (1937), Moderne Algebra.

• Welsh, D. J. A. (1976), Matroid Theory, L.M.S. Monographs 8, Academic Press, ISBN 0-12-744050-X, Zbl0343.05002.

• White, Neil, ed. (1986), Theory of Matroids, Encyclopedia of Mathematics and its Applications 26, Cam-bridge: Cambridge University Press, ISBN 978-0-521-30937-0, Zbl 0579.00001.

• White, Neil, ed. (1987), Combinatorial geometries, Encyclopedia of Mathematics and its Applications 29,Cambridge: Cambridge University Press, ISBN 0-521-33339-3, Zbl 0626.00007

• White, Neil, ed. (1992), Matroid Applications, Encyclopedia of Mathematics and its Applications 40, Cam-bridge: Cambridge University Press, ISBN 978-0-521-38165-9, Zbl 0742.00052.

• Whitney, Hassler (1935), “On the abstract properties of linear dependence”, American Journal of Mathe-matics (The Johns Hopkins University Press) 57 (3): 509–533, doi:10.2307/2371182, JSTOR 2371182, MR1507091. Reprinted in Kung (1986), pp. 55–79.

• Whittle, Geoff (1995), “A characterization of the matroids representable over GF(3) and the rationals” (PDF),Journal of Combinatorial Theory Series B 65 (2): 222–261, doi:10.1006/jctb.1995.1052.

33.13 External links• Hazewinkel, Michiel, ed. (2001), “Matroid”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-

4

• Kingan, Sandra : Matroid theory. A large bibliography of matroid papers, matroid software, and links.

• Locke, S. C. : Greedy Algorithms.

• Pagano, Steven R. : Matroids and Signed Graphs.

• Mark Hubenthal: A Brief Look At Matroids (pdf) (contain proofs for staments of this article)

• James Oxley : What is a matroid?

• Neil White : Matroid Applications

Chapter 34

Maximum coverage problem

The maximum coverage problem is a classical question in computer science, computational complexity theory, andoperations research. It is a problem that is widely taught in approximation algorithms.As input you are given several sets and a number k . The sets may have some elements in common. You must selectat most k of these sets such that the maximum number of elements are covered, i.e. the union of the selected setshas maximal size.Formally, (unweighted) Maximum Coverage

Instance: A number k and a collection of sets S = S1, S2, . . . , Sm .Objective: Find a subset S′ ⊆ S of sets, such that

∣∣∣S′∣∣∣ ≤ k and the number of covered elements∣∣∪

Si∈S′ Si

∣∣ is maximized.

The maximum coverage problem is NP-hard, and cannot be approximated within 1− 1e+o(1) ≈ 0.632 under standard

assumptions. This result essentially matches the approximation ratio achieved by the generic greedy algorithm usedfor maximization of submodular functions with a cardinality constraint.[1]

34.1 ILP formulation

The maximum coverage problem can be formulated as the following integer linear program.

34.2 Greedy algorithm

The greedy algorithm for maximum coverage chooses sets according to one rule: at each stage, choose a set whichcontains the largest number of uncovered elements. It can be shown that this algorithm achieves an approximationratio of 1− 1

e .[2] Inapproximability results show that the greedy algorithm is essentially the best-possible polynomialtime approximation algorithm for maximum coverage.[3]

34.3 Known extensions

The inapproximability results apply to all extensions of the maximum coverage problem since they hold the maximumcoverage problem as a special case.

34.4 Weighted version

In the weighted version every element ej has a weight w(ej) . The task is to find a maximum coverage which hasmaximum weight. The basic version is a special case when all weights are 1 .

127

128 CHAPTER 34. MAXIMUM COVERAGE PROBLEM

maximize∑

e∈E w(ej) · yj . (maximizing the weighted sum of covered elements).subject to

∑xi ≤ k ; (no more than k sets are selected).∑

ej∈Sixi ≥ yj ; (if yj ≥ 0 then at least one set ej ∈ Si is selected).

yj ∈ 0, 1 ; (if yj = 1 then ej is covered)xi ∈ 0, 1 (if xi = 1 then Si is selected for the cover).

The greedy algorithm for the weighted maximum coverage at each stage chooses a set which contains the maximumweight of uncovered elements. This algorithm achieves an approximation ratio of 1− 1

e .[1]

34.5 Budgeted maximum coverage

In the budgeted maximum coverage version, not only does every element ej have a weight w(ej) , but also every setSi has a cost c(Si) . Instead of k that limits the number of sets in the cover a budget B is given. This budget B limitsthe weight of the cover that can be chosen.

maximize∑

e∈E w(ej) · yj . (maximizing the weighted sum of covered elements).subject to

∑c(Si) · xi ≤ B ; (the cost of the selected sets cannot exceed B ).∑

ej∈Sixi ≥ yj ; (if yj ≥ 0 then at least one set ej ∈ Si is selected).

yj ∈ 0, 1 ; (if yj = 1 then ej is covered)xi ∈ 0, 1 (if xi = 1 then Si is selected for the cover).

A greedy algorithm will no longer produce solutions with a performance guarantee. Namely, the worst case behaviorof this algorithm might be very far from the optimal solution. The approximation algorithm is extended by thefollowing way. First, after finding a solution using the greedy algorithm, return the better of the greedy algorithm’ssolution and the set of largest weight. Call this algorithm the modified greedy algorithm. Second, starting withall possible families of sets of sizes from one to (at least) three, augment these solutions with the modified greedyalgorithm. Third, return the best out of all augmented solutions. This algorithm achieves an approximation ratio of1− 1/e . This is the best possible approximation ratio unless NP ⊆ DTIME(nO(log log n)) .[4]

34.6 Generalized maximum coverage

In the generalized maximum coverage version every set Si has a cost c(Si) , element ej has a different weight andcost depending on which set covers it. Namely, if ej is covered by set Si the weight of ej is wi(ej) and its cost isci(ej) . A budget B is given for the total cost of the solution.

maximize∑

e∈E,Siwi(ej)·yij . (maximizing the weighted sum of covered elements in the sets in which

they are covered).subject to

∑ci(ej) · yij +

∑c(Si) · xi ≤ B ; (the cost of the selected sets cannot exceed B ).∑

i yij ≤ 1 ; (element ej = 1 can only be covered by at most one set).∑Si

xi ≥ yij ; (if yj ≥ 0 then at least one set ej ∈ Si is selected).yij ∈ 0, 1 ; (if yij = 1 then ej is covered by set Si )xi ∈ 0, 1 (if xi = 1 then Si is selected for the cover).

34.6.1 Generalized maximum coverage algorithm

The algorithm uses the concept of residual cost/weight. The residual cost/weight is measured against a tentativesolution and it is the difference of the cost/weight from the cost/weight gained by a tentative solution.The algorithm has several stages. First, find a solution using greedy algorithm. In each iteration of the greedyalgorithm the tentative solution is added the set which contains the maximum residual weight of elements divided bythe residual cost of these elements along with the residual cost of the set. Second, compare the solution gained by thefirst step to the best solution which uses a small number of sets. Third, return the best out of all examined solutions.This algorithm achieves an approximation ratio of 1− 1/e− o(1) .[5]

34.7. RELATED PROBLEMS 129

34.7 Related problems• Set cover problem is to cover all elements with as few sets as possible.

34.8 Notes[1] G. L. Nemhauser, L. A. Wolsey and M. L. Fisher. An analysis of approximations for maximizing submodular set functions

I, Mathematical Programming 14 (1978), 265–294

[2] Hochbaum, Dorit S. (1997). “Approximating Covering and Packing Problems: Set Cover, Vertex Cover, Independent Set,and Related Problems”. In Hochbaum, Dorit S.Approximation Algorithms for NP-Hard Problems. Boston: PWS PublishingCompany. pp. 94–143. ISBN 053494968-1.

[3] Feige, Uriel (July 1998). “A Threshold of ln n for Approximating Set Cover”. Journal of the ACM 45 (4) (New York, NY,USA: Association for Computing Machinery). pp. 634–652. doi:10.1145/285055.285059. ISSN 0004-5411.

[4] Khuller, S., Moss, A., and Naor, J. 1999. The budgeted maximum coverage problem. Inf. Process. Lett. 70, 1 (Apr.1999), 39-45.

[5] Cohen, R. and Katzir, L. 2008. The Generalized Maximum Coverage Problem. Inf. Process. Lett. 108, 1 (Sep. 2008),15-22.

34.9 References• Vazirani, Vijay V. (2001). Approximation Algorithms. Springer-Verlag. ISBN 3-540-65367-8.

34.10 External links

Chapter 35

Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras.The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest σ-algebracontaining G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini’s theorem.

35.1 Definition of a monotone class

A monotone class in a set R is a collection M of subsets of R which contains R and is closed under countablemonotone unions and intersections, i.e. if Ai ∈ M and A1 ⊂ A2 ⊂ . . . then ∪∞i=1Ai ∈ M , and similarly forintersections of decreasing sequences of sets.

35.2 Monotone class theorem for sets

35.2.1 Statement

Let G be an algebra of sets and define M(G) to be the smallest monotone class containing G. Then M(G) is preciselythe σ-algebra generated by G, i.e. σ(G) = M(G)

35.3 Monotone class theorem for functions

35.3.1 Statement

LetA be a π-system that contains Ω and letH be a collection of functions from Ω to R with the following properties:(1) If A ∈ A , then 1A ∈ H(2) If f, g ∈ H , then f + g and cf ∈ H for any real number c(3) If fn ∈ H is a sequence of non-negative functions that increase to a bounded function f , then f ∈ HThenH contains all bounded functions that are measurable with respect to σ(A) , the sigma-algebra generated byA

35.3.2 Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples. [1]

The assumption Ω ∈ A , (2) and (3) imply that G = A : 1A ∈ H is a λ-system. By (1) and the π − λ theorem,σ(A) ⊂ G . (2) implies H contains all simple functions, and then (3) implies that H contains all bounded functionsmeasurable with respect to σ(A) .

130

35.4. RESULTS AND APPLICATIONS 131

35.4 Results and Applications

As a corollary, if G is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring ofG.By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a sigma-algebra.The monotone class theorem for functions can be a powerful tool that allows statements about particularly simpleclasses of functions to be generalized to arbitrary bounded and measurable functions.

35.5 References[1] Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 100. ISBN 978-

0521765398.

35.6 See also

This article was advanced during a Wikipedia course held at Duke University, which can be found here: Wikipediaand Its Ancestors

Chapter 36

Near polygon

A dense near polygon with diameter d = 2

In mathematics, a near polygon is an incidence geometry introduced by Ernest E. Shult and Arthur Yanushka in1980.[1] Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Eu-clidean spaces and a class of point-line geometries which they called near polygons. These structures generalise the

132

36.1. DEFINITION 133

notion of generalized polygon as every generalized 2n-gon is a near 2n-gon of a particular kind. These structureswere extensively studied and connection between them and dual polar space [2] was shown in 1980s and early 1990s.Some sporadic simple groups, for example the Hall-Janko group and the Mathieu groups, act as automorphism groupsof near polygons.

36.1 Definition

A near 2d-gon is an incidence structure ( P,L, I ), where P is the set of points, L is the set of lines and I ⊆ P × Lis the incidence relation, such that:

• The maximum distance between two points (the so-called diameter) is d.• For every point x and every line L there exists a unique point on L which is nearest to x .

Note that the distance are measure in the collinearity graph of points, i.e., the graph formed by taking points asvertices and joining a pair of vertices if they are incident with a common line. We can also give an alternate graphtheoretic definition, a near 2d-gon is a connected graph of finite diameter d with the property that for every vertex xand every maximal clique M there exists a unique vertex x' in M nearest to x. The maximal cliques of such a graphcorrespond to the lines in the incidence structure definition. A near 0-gon (d = 0) is a single point while a near 2-gon(d = 1) is just a single line, i.e., a complete graph. A near quadrangle (d = 2) is same as a (possibly degenerate)generalized quadrangle. In fact, it can be shown that every generalized 2d-gon is a near 2d-gon that satisfies thefollowing two additional conditions:

• Every point is incident with at least two lines.• For every two points x, y at distance i < d, there exists a unique neighbour of y at distance i − 1 from x.

A near polygon is called dense if every line is incident with at least three points and if every two points at distancetwo have at least two common neighbours. It is said to have order (s, t) if every line is incident with precisely s +1 points and every point is incident with precisely t + 1 lines. Dense near polygons have a rich theory and severalclasses of them (like the slim dense near polygons) have been completely classified.[3]

36.2 Examples• All connected bipartite graphs are near polygons. In fact any near polygon which has precisely two points per

line must be a connected bipartite graph.• All finite generalized polygons except the projective planes.• All dual polar spaces.• The Hall–Janko near octagon[4] associated with the Hall–Janko group. It can be constructed by choosing the

conjugacy class of 315 central involutions of the Hall-Janko group as points and lines as three element subsetsx,y,xy whenever x and y commute.

• The M24 near hexagon related to the Mathieu group M24.[5]

• Take the partitions of 1, 2,..., 2n+2 into n+1 2-subsets as points and the partitions into n 2-subsets and one4-subset as lines. A point is incident to a line if as a partition it is a refinement of the line. This gives us a near2n-gon with three points on each line, usually denoted as H . Its full automorphism group is S₂ ₊₂.[5]

36.3 Regular near polygons

A finite near 2d -gon S is called regular if it has an order (s, t) and if there exist constants ti, i ∈ 1, . . . , d , suchthat for every two points x and y at distance i , there are precisely ti + 1 lines through y containing a (necessarilyunique) point at distance i − 1 from x . It turns out that regular near 2d -gons are precisely those near 2d -gonswhose point graph is a distance-regular graph. A generalized 2d -gon of order (s, t) is a regular near 2d -gon withparameters t1 = 0, t2 = 0, . . . , td = t

134 CHAPTER 36. NEAR POLYGON

36.4 See also• Finite geometry

• Polar space

• Partial linear space

• Association scheme

• Hall–Janko graph

36.5 Notes[1] Shult, Ernest; Yanushka, Arthur. “Near n-gons and line systems”.

[2] Cameron, Peter J. “Dual polar spaces”.

[3] De Bruyn, Bart. Near Polygons

[4] http://www.win.tue.nl/~aeb/graphs/HJ315.html

[5] http://oai.cwi.nl/oai/asset/1493/1493A.pdf

36.6 References• Shult, Ernest; Yanushka, Arthur (1980), “Near n-gons and line systems”,Geom. Dedicata 9: 1––72, doi:10.1007/BF00156473,

MR 566437.

• Cameron, Peter J. (1982), “Dual polar spaces”, Geom. Dedicata 12: 75–85, doi:10.1007/bf00147332, MR645040.

• Brouwer, A.E.; Cohen, A. M.; Wilbrink, H. A.; Hall, J. J. (1994), “Near polygons and Fischer spaces”, Geom.Dedicata 49 (3): 349–368, doi:10.1007/BF01264034

• De Bruyn, Bart (2006),Near Polygons, Frontiers in Mathematics, Birkhäuser Verlag, doi:10.1007/978-3-7643-7553-9, ISBN 3-7643-7552-3, MR 2227553.

• Shult, Ernest E. (2011), Points and Lines, Universitext, Springer, doi:10.1007/978-3-642-15627-4, ISBN 978-3-642-15626-7.

• Brouwer, A.E.; Cohen, A.M. (1989), Distance Regular Graphs, Berlin, New York: Springer-Verlag., ISBN3-540-50619-5, MR 1002568.

• Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes 13, London: Queen Mary andWestfield College School of Mathematical Sciences, MR 1153019

Chapter 37

Nerve (category theory)

In category theory, a discipline within mathematics, the nerve N(C) of a small category C is a simplicial set con-structed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space,called the classifying space of the category C. These closely related objects can provide information about somefamiliar and useful categories using algebraic topology, most often homotopy theory.

37.1 Motivation

The nerve of a category is often used to construct topological versions of moduli spaces. If X is an object of C,its moduli space should somehow encode all objects isomorphic to X and keep track of the various isomorphismsbetween all of these objects in that category. This can become rather complicated, especially if the objects havemany non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicialsets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups π (N(C)).One hopes that the answers to such questions provide interesting information about the original category C, or aboutrelated categories.The notion of nerve is a direct generalization of the classical notion of classifying space of a discrete group; see belowfor details.

37.2 Construction

Let C be a small category. It is easy to define the sets N(C)k for small k, which leads to the general definition. Inparticular, there is a 0-simplex of N(C) for each object of C. There is a 1-simplex for each morphism f : x → y in C.Now suppose that f: x → y and g : y → z are morphisms in C. Then we also have their composition gf : x → z.The diagram suggests our course of action: add a 2-simplex for this commutative triangle. Every 2-simplex of N(C)comes from a pair of composable morphisms in this way. Note that the addition of these 2-simplices does not eraseor otherwise disregard morphisms obtained by composition, it merely remembers that that is how they arise.In general, N(C)k consists of the k-tuples of composable morphisms

A0 → A1 → A2 → · · · → Ak−1 → Ak

of C. To complete the definition of N(C) as a simplicial set, we must also specify the face and degeneracy maps.These are also provided to us by the structure of C as a category. The face maps

di : N(C)k → N(C)k−1

are given by composition of morphisms at the ith object (or removing the ith object from the sequence, when i is 0or k).[1] This means that di sends the k-tuple

135

136 CHAPTER 37. NERVE (CATEGORY THEORY)

A 2-simplex.

A0 → · · · → Ai−1 → Ai → Ai+1 → · · · → Ak

to the (k − 1)-tuple

A0 → · · · → Ai−1 → Ai+1 → · · · → Ak.

That is, the map di composes the morphisms Ai₋₁ → Ai and Ai → Ai₊₁ into the morphism Ai₋₁ → Ai₊₁, yielding a (k− 1)-tuple for every k-tuple.Similarly, the degeneracy maps

si : N(C)k → N(C)k+1

are given by inserting an identity morphism at the object Ai.

37.3. EXAMPLES 137

Recall that simplicial sets may also be regarded as functors Δop → Set, where Δ is the category of totally orderedfinite sets and order-preserving morphisms. Every partially ordered set P yields a (small) category i(P) with objectsthe elements of P and with a unique morphism from p to q whenever p ≤ q in P. We thus obtain a functor i from thecategory Δ to the category of small categories. We can now describe the nerve of the category C as the functor Δop

→ Set

N(C)(?) = Fun(i(?), C).

This description of the nerve makes functoriality quite transparent; for example, a functor between small categories Cand D induces a map of simplicial sets N(C) → N(D). Moreover a natural transformation between two such functorsinduces a homotopy between the induced maps. This observation can be regarded as the beginning of one of theprinciples of higher category theory. It follows that adjoint functors induce homotopy equivalences. In particular, ifC has an initial or final object, its nerve is contractible.

37.3 Examples

The primordial example is the classifying space of a discrete group G. We regard G as a category with one objectwhose endomorphisms are the elements of G. Then the k-simplices of N(G) are just k-tuples of elements of G. Theface maps act by multiplication, and the degeneracy maps act by insertion of the identity element. If G is the groupwith two elements, then there is exactly one nondegenerate k-simplex for each nonnegative integer k, correspondingto the unique k-tuple of elements of G containing no identities. After passing to the geometric realization, it is nothard to see that this k-tuple can be identified with the unique k-cell in the usual CW structure on infinite-dimensionalreal projective space. The latter is the most popular model for the classifying space of the group with two elements.See (Segal 1968) for further details and the relationship of the above to Milnor’s join construction of BG.

37.3.1 Most spaces are classifying spaces

It is well known that every “reasonable” topological space is homeomorphic to the classifying space of a small category.Here, “reasonable” means that the space in question is the geometric realization of a simplicial set. This is obviouslya necessary condition; it is perhaps surprising that it is also sufficient. Indeed, let X be the geometric realization ofa simplicial set K. The set of simplices in K is partially ordered, by the relation x ≤ y if and only if x is a face ofy. Of course, we may consider this partially ordered set as a category. The nerve of this category is the barycentricsubdivision of K, and thus its realization is homeomorphic to X, because X is the realization of K by hypothesis andbarycentric subdivision does not change the homeomorphism type of the realization.

37.3.2 The nerve of an open covering

Main article: Nerve of an open covering

IfX is a topological space with open coverUi, the nerve of the cover is obtained from the above definitions by replacingthe cover with the category obtained by regarding the cover as a partially ordered set with relation that of set inclusion.Note that the realization of this nerve is not generally homeomorphic to X (or even homotopy equivalent).

37.3.3 A moduli example

One can use the nerve construction to recover mapping spaces, and even get “higher-homotopical” information aboutmaps. LetD be a category, and letX and Y be objects ofD. One is often interested in computing the set of morphismsX→Y. We can use a nerve construction to recover this set. LetC =C(X,Y) be the category whose objects are diagrams

X ←− U −→ V ←− Y

such that the morphisms U → X and Y → V are isomorphisms in D. Morphisms in C(X, Y) are diagrams of thefollowing shape:

138 CHAPTER 37. NERVE (CATEGORY THEORY)

Here, the indicated maps are to be isomorphisms or identities. The nerve of C(X, Y) is the moduli space of maps X→ Y. In the appropriate model category setting, this moduli space is weak homotopy equivalent to the simplicial setof morphisms of D from X to Y.

37.4 References[1] The ith face of the simplex is then the one missing the ith vertex.

• Blanc, D., W. G. Dwyer, and P.G. Goerss. “The realization space of a Π -algebra: a moduli problem inalgebraic topology.” Topology 43 (2004), no. 4, 857–892.

• Goerss, P. G., and M. J. Hopkins. "Moduli spaces of commutative ring spectra.” Structured ring spectra, 151–200, London Math. Soc. Lecture Note Ser., 315, Cambridge Univ. Press, Cambridge, 2004.

• Segal, Graeme. “Classifying spaces and spectral sequences.” Inst. Hautes Études Sci. Publ. Math. No. 34(1968) 105–112.

• Nerve in nLab

Chapter 38

Nerve of a covering

In mathematics, the nerve of an open covering is a construction in topology, of an abstract simplicial complex froman open covering of a topological space X.The notion of nerve was introduced by Pavel Alexandrov.[1]

Given an index set I, and open sets Ui contained in X, the nerve N is the set of finite subsets of I defined as follows:

• a finite set J ⊆ I belongs to N if and only if the intersection of the Ui whose subindices are in J is non-empty.That is, if and only if

∩j∈J

Uj = ∅.

Obviously, if J belongs to N, then any of its subsets is also in N. Therefore N is an abstract simplicial complex.In general, the complex N need not reflect the topology of X accurately. For example we can cover any n-spherewith two contractible sets U and V, in such a way that N is a 1-simplex. However, if we also insist that the open setscorresponding to every intersection indexed by a set in N is also contractible, the situation changes. This means forinstance that a circle covered by three open arcs, intersecting in pairs in one arc, is modelled by a homeomorphiccomplex, the geometrical realization of N.

38.1 Notes[1] Paul Alexandroff Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen An-

schauung, — Mathematische Annalen 98 (1928), стр. 617—635.

38.2 References• Samuel Eilenberg and Norman Steenrod: Foundations of Algebraic Topology, Princeton University Press, 1952,

p. 234.

139

Chapter 39

Noncrossing partition

In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (amongother things) its application to the theory of free probability. The set of all noncrossing partitions is one of many setsenumerated by the Catalan numbers. The number of noncrossing partitions of an n-element set with k blocks is foundin the Narayana number triangle.

39.1 Definition

A partition of a set S is a pairwise disjoint set of non-empty subsets, called “parts” or “blocks”, whose union is all ofS. Consider a finite set that is linearly ordered, or (equivalently, for purposes of this definition) arranged in a cyclicorder like the vertices of a regular n-gon. No generality is lost by taking this set to be S = 1, ..., n . A noncrossingpartition of S is a partition in which no two blocks “cross” each other, i.e., if a and b belong to one block and x andy to another, they are not arranged in the order a x b y. If one draws an arch based at a and b, and another arch basedat x and y, then the two arches cross each other if the order is a x b y but not if it is a x y b or a b x y. In the lattertwo orders the partition a, b , x, y is noncrossing.Equivalently, if we label the vertices of a regular n-gon with the numbers 1 through n, the convex hulls of differentblocks of the partition are disjoint from each other, i.e., they also do not “cross” each other. The set of all non-crossingpartitions of S are denoted NC(S) . There is an obvious order isomorphism between NC(S1) and NC(S2) for twofinite sets S1, S2 with the same size. That is, NC(S) depends essentially only on the size of S and we denote byNC(n) the non-crossing partitions on any set of size n.

39.2 Lattice structure

Like the set of all partitions of the set 1, ..., n , the set of all noncrossing partitions is a lattice when partiallyordered by saying that a finer partition is “less than” a coarser partition. However, although it is a subset of the latticeof all partitions, it is not a sublattice of the lattice of all partitions, because the join operations do not agree. In otherwords, the finest partition that is coarser than both of two noncrossing partitions is not always the finest noncrossingpartition that is coarser than both of them.Unlike the lattice of all partitions of the set, the lattice of all noncrossing partitions of a set is self-dual, i.e., it is order-isomorphic to the lattice that results from inverting the partial order (“turning it upside-down”). This can be seen byobserving that each noncrossing partition has a complement. Indeed, every interval within this lattice is self-dual.

39.3 Role in free probability theory

The lattice of noncrossing partitions plays the same role in defining free cumulants in free probability theory that isplayed by the lattice of all partitions in defining joint cumulants in classical probability theory. To be more precise,let (A, ϕ) be a non-commutative probability space (See free probability for terminology.), a ∈ A a non-commutativerandom variable with free cumulants (kn)n∈N . Then

140

39.4. REFERENCES 141

ϕ(an) =∑

π∈NC(n)

∏j

kNj(π)j

whereNj(π) denotes the number of blocks of length j in the non-crossing partition π . That is, the moments of a non-commutative random variable can be expressed as a sum of free cumulants over the sum non-crossing partitions. Thisis the free analogue of the moment-cumulant formula in classical probability. See also Wigner semicircle distribution.

39.4 References• Germain Kreweras, “Sur les partitions non croisées d'un cycle”, Discrete Mathematics, volume 1, number 4,

pages 333–350, 1972.

• Rodica Simion, “Noncrossing partitions”, Discrete Mathematics, volume 217, numbers 1–3, pages 367–409,April 2000.

• Roland Speicher, “Free probability and noncrossing partitions”, Séminaire Lotharingien de Combinatoire, B39c(1997), 38 pages, 1997

142 CHAPTER 39. NONCROSSING PARTITION

There are 42 noncrossing and 10 crossing partitions of a 5-element set

39.4. REFERENCES 143

The 14 noncrossing partitions of a 4-element set ordered in a Hasse diagram

Chapter 40

Partition of a set

For the partition calculus of sets, see infinitary combinatorics.In mathematics, a partition of a set is a grouping of the set’s elements into non-empty subsets, in such a way that

A set of stamps partitioned into bundles: No stamp is in two bundles, and no bundle is empty.

every element is included in one and only one of the subsets.

144

40.1. DEFINITION 145

40.1 Definition

A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of thesesubsets[1] (i.e., X is a disjoint union of the subsets).Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold:[2]

1. P does not contain the empty set.

2. The union of the sets in P is equal to X. (The sets in P are said to cover X.)

3. The intersection of any two distinct sets in P is empty. (We say the elements of P are pairwise disjoint.)

In mathematical notation, these conditions can be represented as

1. ∅ /∈ P

2.∪

A∈P A = X

3. if A,B ∈ P and A = B then A ∩B = ∅ ,

where ∅ is the empty set.The sets in P are called the blocks, parts or cells of the partition.[3]

The rank of P is |X| − |P|, if X is finite.

40.2 Examples• Every singleton set x has exactly one partition, namely x .

• For any nonempty set X, P = X is a partition of X, called the trivial partition.

• For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U,namely, A, U−A.

• The set 1, 2, 3 has these five partitions:

• 1, 2, 3 , sometimes written 1|2|3.• 1, 2, 3 , or 12|3.• 1, 3, 2 , or 13|2.• 1, 2, 3 , or 1|23.• 1, 2, 3 , or 123 (in contexts where there will be no confusion with the number).

• The following are not partitions of 1, 2, 3 :

• , 1, 3, 2 is not a partition (of any set) because one of its elements is the empty set.• 1, 2, 2, 3 is not a partition (of any set) because the element 2 is contained in more than one block.• 1, 2 is not a partition of 1, 2, 3 because none of its blocks contains 3; however, it is a partition

of 1, 2.

40.3 Partitions and equivalence relations

For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from anypartition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the samepart in P. Thus the notions of equivalence relation and partition are essentially equivalent.[4]

The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly oneelement from each part of the partition. This implies that given an equivalence relation on a set one can select acanonical representative element from every equivalence class.

146 CHAPTER 40. PARTITION OF A SET

40.4 Refinement of partitions

A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarserthan α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentationof ρ. In that case, it is written that α ≤ ρ.This finer-than relation on the set of partitions of X is a partial order (so the notation "≤" is appropriate). Each setof elements has a least upper bound and a greatest lower bound, so that it forms a lattice, and more specifically (forpartitions of a finite set) it is a geometric lattice.[5] The partition lattice of a 4-element set has 15 elements and isdepicted in the Hasse diagram on the left.Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a finite set cor-responds to a matroid in which the base set of the matroid consists of the atoms of the lattice, the partitions withn − 2 singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of acomplete graph. The matroid closure of a set of atomic partitions is the finest common coarsening of them all; ingraph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of thesubgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the graphic matroidof the complete graph.Another example illustrates the refining of partitions from the perspective of equivalence relations. If D is the set ofcards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalenceclasses: the sets red cards and black cards. The 2-part partition corresponding to ~C has a refinement that yieldsthe same-suit-as relation ~S, which has the four equivalence classes spades, diamonds, hearts, and clubs.

40.5 Noncrossing partitions

A partition of the set N = 1, 2, ..., n with corresponding equivalence relation ~ is noncrossing provided that forany two 'cells’ C1 and C2, either all the elements in C1 are < than all the elements in C2 or they are all > than all theelements in C2. In other words: given distinct numbers a, b, c in N, with a < b < c, if a ~ c (they both are in a cellcalled C), it follows that also a ~ b and b ~ c, that is b is also in C. The lattice of noncrossing partitions of a finite sethas recently taken on importance because of its role in free probability theory. These form a subset of the lattice ofall partitions, but not a sublattice, since the join operations of the two lattices do not agree.

40.6 Counting partitions

The total number of partitions of an n-element set is the Bell number Bn. The first several Bell numbers are B0 =1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, and B6 = 203 (sequence A000110 in OEIS). Bell numbers satisfy therecursion

Bn+1 =n∑

k=0

(n

k

)Bk

and have the exponential generating function

∞∑n=0

Bn

n!zn = ee

z−1.

The Bell numbers may also be computed using the Bell triangle in which the first value in each row is copied fromthe end of the previous row, and subsequent values are computed by adding the two numbers to the left and above leftof each position. The Bell numbers are repeated along both sides of this triangle. The numbers within the trianglecount partitions in which a given element is the largest singleton.The number of partitions of an n-element set into exactly k nonempty parts is the Stirling number of the second kindS(n, k).The number of noncrossing partitions of an n-element set is the Catalan number Cn, given by

40.7. SEE ALSO 147

Cn =1

n+ 1

(2n

n

).

40.7 See also• Exact cover

• Cluster analysis

• Weak ordering (ordered set partition)

• Equivalence relation

• Partial equivalence relation

• Partition refinement

• List of partition topics

• Lamination (topology)

• Rhyme schemes by set partition

40.8 Notes[1] Naive Set Theory (1960). Halmos, Paul R. Springer. p. 28. ISBN 9780387900926.

[2] Lucas, John F. (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 187. ISBN 9780912675732.

[3] Brualdi, pp. 44–45

[4] Schechter, p. 54

[5] Birkhoff, Garrett (1995), Lattice Theory, Colloquium Publications 25 (3rd ed.), American Mathematical Society, p. 95,ISBN 9780821810255.

40.9 References• Brualdi, Richard A. (2004). Introductory Combinatorics (4th ed.). Pearson Prentice Hall. ISBN 0-13-100119-

1.

• Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.

148 CHAPTER 40. PARTITION OF A SET

The 52 partitions of a set with 5 elements

40.9. REFERENCES 149

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

The traditional Japanese symbols for the chapters of the Tale of Genji are based on the 52 ways of partitioning five elements.

150 CHAPTER 40. PARTITION OF A SET

Partitions of a 4-set ordered by refinement

40.9. REFERENCES 151

Construction of the Bell triangle

Chapter 41

Partition regularity

In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets.Given a set X , a collection of subsets S ⊂ P(X) is called partition regular if every set A in the collection has theproperty that, no matter how A is partitioned into finitely many subsets, at least one of the subsets will also belong tothe collection. That is, for any A ∈ S , and any finite partition A = C1 ∪ C2 ∪ · · · ∪ Cn , there exists an i ≤ n, suchthat Ci belongs to S . Ramsey theory is sometimes characterized as the study of which collections S are partitionregular.

41.1 Examples• the collection of all infinite subsets of an infinite setX is a prototypical example. In this case partition regularity

asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.)

• sets with positive upper density inN : the upper density d(A) ofA ⊂ N is defined as d(A) = lim supn→∞|1,2,...,n∩A|

n .

• For any ultrafilter U on a set X , U is partition regular. If U ∋ A =∪n

1 Ci , then for exactly one i is Ci ∈ U .

• sets of recurrence: a set R of integers is called a set of recurrence if for any measure preserving transformationT of the probability space (Ω, β, μ) and A ∈ β of positive measure there is a nonzero n ∈ R so thatµ(A ∩ TnA) > 0 .

• Call a subset of natural numbers a.p.-rich if it contains arbitrarily long arithmetic progressions. Then thecollection of a.p.-rich subsets is partition regular (Van der Waerden, 1927).

• Let [A]n be the set of all n-subsets of A ⊂ N . Let Sn =∪

A⊂N[A]n . For each n, Sn is partition regular.

(Ramsey, 1930).

• For each infinite cardinal κ , the collection of stationary sets of κ is partition regular. More is true: if S isstationary and S =

∪α<λ Sα for some λ < κ , then some Sα is stationary.

• the collection of ∆ -sets: A ⊂ N is a ∆ -set if A contains the set of differences sm−sn : m,n ∈ N, n < mfor some sequence ⟨sn⟩ωn=1 .

• the set of barriers on N : call a collection B of finite subsets of N a barrier if:

• ∀X,Y ∈ B, X ⊂ Y and• for all infinite I ⊂ ∪B , there is some X ∈ B such that the elements of X are the smallest elements of I;i.e. X ⊂ I and ∀i ∈ I \X,∀x ∈ X,x < i .

This generalizes Ramsey’s theorem, as each [A]n is a barrier. (Nash-Williams, 1965)

152

41.2. REFERENCES 153

• finite products of infinite trees (Halpern–Läuchli, 1966)

• piecewise syndetic sets (Brown, 1968)

• Call a subset of natural numbers i.p.-rich if it contains arbitrarily large finite sets together with all their finitesums. Then the collection of i.p.-rich subsets is partition regular (Folkman–Rado–Sanders, 1968).

• (m, p, c)-sets (Deuber, 1973)

• IP sets (Hindman, 1974, see also Hindman, Strauss, 1998)

• MTk sets for each k, i.e. k-tuples of finite sums (Milliken–Taylor, 1975)

• central sets; i.e. the members of any minimal idempotent in βN , the Stone–Čech compactification of theintegers. (Furstenberg, 1981, see also Hindman, Strauss, 1998)

41.2 References1. Vitaly Bergelson, N. Hindman Partition regular structures contained in large sets are abundant J. Comb. Theory

(Series A) 93 (2001), 18–36.

2. T. Brown, An interesting combinatorial method in the theory of locally finite semigroups, Pacific J. Math. 36,no. 2 (1971), 285–289.

3. W. Deuber, Mathematische Zeitschrift 133, (1973) 109–123

4. N. Hindman, Finite sums from sequences within cells of a partition of N, J. Combinatorial Theory (Series A)17 (1974) 1–11.

5. C.St.J.A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965),33–39.

6. N. Hindman, D. Strauss, Algebra in the Stone–Čech compactification, De Gruyter, 1998

7. J.Sanders, A Generalization of Schur’s Theorem, Doctoral Dissertation, Yale University, 1968.

Chapter 42

Pi system

In mathematics, a π-system (or pi-system) on a set Ω is a collection P of certain subsets of Ω, such that

• P is non-empty.

• A ∩ B ∈ P whenever A and B are in P.

That is, P is a non-empty family of subsets of Ω that is closed under finite intersections. The importance of π-systemsarise from the fact that if two probability measures agree on a π-system, then they agree on the σ-algebra generatedby that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they holdfor the generated σ-algebra as well. This is the case whenever the collection of subsets for which the property holdsis a λ-system. π-systems are also useful for checking independence of random variables.This is desirable because in practice, π-systems are often simpler to work with than σ-algebras. For example, it maybe awkward to work with σ-algebras generated by infinitely many sets σ(E1, E2, . . .) . So instead we may examinethe union of all σ-algebras generated by finitely many sets

∪n σ(E1, . . . , En) . This forms a π-system that generates

the desired σ-algebra. Another example is the collection of all interval subsets of the real line, along with the emptyset, which is a π-system that generates the very important Borel σ-algebra of subsets of the real line.

42.1 Examples

• ∀a, b ∈ R , the intervals (−∞, a] form a π-system, and the intervals (a, b] form a π-system, if the empty setis also included.

• The topology (collection of open subsets) of any topological space is a π-system.

• For any collection Σ of subsets of Ω, there exists a π-system IΣ which is the unique smallest π-system of Ω tocontain every element of Σ, and is called the π-system generated by Σ.

• For any measurable function f : Ω → R , the set If =f−1 ((−∞, x]) : x ∈ R

defines a π-system, and is

called the π-system generated by f. (Alternatively,f−1 ((a, b]) : a, b ∈ R, a < b

∪∅ defines a π-system

generated by f .)

• If P1 and P2 are π-systems for Ω1 and Ω2, respectively, then A1 × A2 : A1 ∈ P1, A2 ∈ P2 is a π-systemfor the product space Ω1×Ω2.

• Any σ-algebra is a π-system.

42.2 Relationship to λ-Systems

A λ-system on Ω is a set D of subsets of Ω, satisfying

154

42.3. Π-SYSTEMS IN PROBABILITY 155

• Ω ∈ D ,

• if A ∈ D then Ac ∈ D ,

• if A1, A2, A3, . . . is a sequence of disjoint subsets in D then ∪∞n=1An ∈ D .

Whilst it is true that any σ-algebra satisfies the properties of being both a π-system and a λ-system, it is not truethat any π-system is a λ-system, and moreover it is not true that any π-system is a σ-algebra. However, a usefulclassification is that any set system which is both a λ-system and a π-system is a σ-algebra. This is used as a step inproving the π-λ theorem.

42.2.1 The π-λ Theorem

Let D be a λ-system, and let I ⊆ D be a π-system contained in D . The π-λ Theorem[1] states that the σ-algebraσ(I) generated by I is contained in D : σ(I) ⊂ D .The π-λ theorem can be used to prove many elementary measure theoretic results. For instance, it is used in provingthe uniqueness claim of the Carathéodory extension theorem for σ-finite measures.[2]

The π-λ theorem is closely related to the monotone class theorem, which provides a similar relationship betweenmonotone classes and algebras, and can be used to derive many of the same results. Since π-systems are simplerclasses than algebras, it can be easier to identify the sets that are in them while, on the other hand, checking whetherthe property under consideration determines a λ-system is often relatively easy. Despite the difference between thetwo theorems, the π-λ theorem is sometimes referred to as the monotone class theorem.[1]

Example

Let μ₁ , μ2 : F → R be two measures on the σ-algebra F, and suppose that F = σ(I) is generated by a π-system I. If

1. μ1(A) = μ2(A), ∀ A ∈ I, and

2. μ1(Ω) = μ2(Ω) < ∞,

then μ₁ = μ2. This is the uniqueness statement of the Carathéodory extension theorem for finite measures. If thisresult does not seem very remarkable, consider the fact that it usually is very difficult or even impossible to fullydescribe every set in the σ-algebra, and so the problem of equating measures would be completely hopeless withoutsuch a tool.Idea of Proof[2] Define the collection of sets

D = A ∈ σ(I) : µ1(A) = µ2(A) .

By the first assumption, μ1 and μ2 agree on I and thus I D. By the second assumption, Ω ∈ D, and it can furtherbe shown that D is a λ-system. It follows from the π-λ theorem that σ(I) D σ(I), and so D = σ(I). That is to say,the measures agree on σ(I).

42.3 π-Systems in Probability

π-systems are more commonly used in the study of probability theory than in the general field of measure theory.This is primarily due to probabilistic notions such as independence, though it may also be a consequence of the factthat the π-λ theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts typically provethe same results via monotone classes, rather than π-systems.

156 CHAPTER 42. PI SYSTEM

42.3.1 Equality in Distribution

The π-λ theorem motivates the common definition of the probability distribution of a random variableX : (Ω,F ,P)→R in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable isdefined as

FX(a) = P [X ≤ a] , a ∈ R

whereas the seemingly more general law of the variable is the probability measure

LX(B) = P[X−1(B)

], B ∈ B(R)

where B(R) is the Borel σ-algebra. We say that the random variables X : (Ω,F ,P) , and Y : (Ω, F , P) → R (ontwo possibly different probability spaces) are equal in distribution (or law), XD

=Y , if they have the same cumulativedistribution functions, FX = FY. The motivation for the definition stems from the observation that if FX = FY, thenthat is exactly to say that LX and LY agree on the π-system (−∞, a] : a ∈ R which generates B(R) , and so bythe example above: LX = LY .A similar result holds for the joint distribution of a random vector. For example, suppose X and Y are two randomvariables defined on the same probability space (Ω,F ,P) , with respectively generated π-systems IX and IY . Thejoint cumulative distribution function of (X,Y) is

FX,Y (a, b) = P [X ≤ a, Y ≤ b] = P[X−1((−∞, a]) ∩ Y −1((−∞, b])

], a, b ∈ R

However, A = X−1((−∞, a]) ∈ IX and B = Y −1((−∞, b]) ∈ IY . Since

IX,Y = A ∩B : A ∈ IX , B ∈ IY

is a π-system generated by the random pair (X,Y), the π-λ theorem is used to show that the joint cumulative distribu-tion function suffices to determine the joint law of (X,Y). In other words, (X,Y) and (W,Z) have the same distributionif and only if they have the same joint cumulative distribution function.In the theory of stochastic processes, two processes (Xt)t∈T , (Yt)t∈T are known to be equal in distribution if andonly if they agree on all finite-dimensional distributions. i.e. for all t1, . . . , tn ∈ T, n ∈ N .

(Xt1 , . . . , Xtn)D=(Yt1 , . . . , Ytn)

The proof of this is another application of the π-λ theorem.[3]

42.3.2 Independent Random Variables

The theory of π-system plays an important role in the probabilistic notion of independence. If X and Y are tworandom variables defined on the same probability space (Ω,F ,P) then the random variables are independent if andonly if their π-systems IX , IY satisfy

P [A ∩B] = P [A]P [B] , ∀A ∈ IX , B ∈ IY ,

which is to say that IX , IY are independent. This actually is a special case of the use of π-systems for determiningthe distribution of (X,Y).

42.4. SEE ALSO 157

Example

Let Z = (Z1, Z2) , where Z1, Z2 ∼ N (0, 1) are iid standard normal random variables. Define the radius andargument (arctan) variables

R =√

Z21 + Z2

2 , Θ = tan−1(Z2/Z1)

Then R and Θ are independent random variables.To prove this, it is sufficient to show that the π-systems IR, IΘ are independent: i.e.

P[R ≤ ρ,Θ ≤ θ] = P[R ≤ ρ]P[Θ ≤ θ] ∀ρ ∈ [0,∞), θ ∈ [0, 2π].

Confirming that this is the case is an exercise in changing variables. Fix ρ ∈ [0,∞), θ ∈ [0, 2π] , then the probabilitycan be expressed as an integral of the probability density function of Z .

P[R ≤ ρ,Θ ≤ θ] =

∫R≤ρ,Θ≤θ

1

2πexp

(−1

2(z21 + z22)

)dz1dz2

=

∫ θ

0

∫ ρ

0

1

2πe−

r2

2 rdrdθ

=

(∫ θ

0

1

2πdθ

)(∫ ρ

0

e−r2

2 rdr

)= P[Θ ≤ θ]P[R ≤ ρ].

42.4 See also• λ-systems

• σ-algebra

• Monotone class theorem

• Probability distribution

• Independence

42.5 Notes[1] Kallenberg, Foundations Of Modern Probability, p.2

[2] Durrett, Probability Theory and Examples, p.404

[3] Kallenberg, Foundations Of Modern probability, p. 48

42.6 References• Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN

0-387-22833-0.

• David Williams (1991). Probability with Martingales. Cambridge University Press. ISBN 0-521-40605-6.

Chapter 43

Polar space

In mathematics, in the field of geometry, a polar space of rank n (n ≥ 3), or projective index n − 1, consists of a setP, conventionally the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms:

• Every subspace, together with its own subspaces, is isomorphic with a projective geometry PG(d, q) with −1≤ d ≤ (n − 1) and q a prime power. By definition, for each subspace the corresponding d is its dimension.

• The intersection of two subspaces is always a subspace.

• For each point p not in a subspace A of dimension of n − 1, there is a unique subspace B of dimension n − 1such that A ∩ B is (n − 2)-dimensional. The points in A ∩ B are exactly the points of A that are in a commonsubspace of dimension 1 with p.

• There are at least two disjoint subspaces of dimension n − 1.

A polar space of rank two is a generalized quadrangle. Finite polar spaces (where P is a finite set) are also studied ascombinatorial objects.

43.1 Examples• In PG(d, q), with d odd and d ≥ 3, the set of all points, with as subspaces the totally isotropic subspaces of an

arbitrary symplectic polarity, forms a polar space of rank (d + 1)/2.

• Let Q be a nonsingular quadric in PG(n, q) with character ω. Then the index of Q will be g = (n + w − 3)/2.The set of all points on the quadric, together with the subspaces on the quadric, forms a polar space of rank g+ 1.

• Let H be a nonsingular Hermitian variety in PG(n, q2). The index of H will be⌊n−12

⌋. The points on H,

together with the subspaces on it, form a polar space of rank⌊n+12

⌋.

43.2 Classification

Jacques Tits proved that a finite polar space of rank at least three, is always isomorphic with one of the three structuresgiven above. This leaves only the problem of classifying generalized quadrangles.

43.3 References• Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes 13, London: Queen Mary and

Westfield College School of Mathematical Sciences, MR 1153019

158

Chapter 44

Pro-simplicial set

In mathematics, a pro-simplicial set is an inverse system of simplicial sets.A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets has finite homotopy groups.Pro-simplicial sets show up in shape theory, in the study of localization and completion in homotopy theory, and inthe study of homotopy properties of schemes (e.g. étale homotopy theory).

44.1 References• Edwards, David A.; Hastings, Harold M. (1980), "Čech theory: its past, present, and future”, The RockyMountain Journal of Mathematics 10 (3): 429–468, doi:10.1216/RMJ-1980-10-3-429, MR 590209.

• Edwards, David A.; Hastings, Harold M. (1976), Čech and Steenrod homotopy theories with applications togeometric topology, Lecture Notes in Mathematics, Vol. 542, Springer-Verlag, Berlin-New York, MR 0428322.

159

Chapter 45

Property B

For the B-property in finite group theory, see B-theorem.

In mathematics, Property B is a certain set theoretic property. Formally, given a finite set X, a collection C of subsetsof X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in Cmeets both Y and Z. The smallest number of sets in a collection of sets of size n such that C does not have PropertyB is denoted by m(n).The property gets its name from mathematician Felix Bernstein, who first introduced the property in 1908.

45.1 Values of m(n)

It is known that m(1) = 1, m(2) = 3, and m(3) = 7 (as can by seen by the following examples); the value of m(4) isnot known, although an upper bound of 23 (Seymour, Toft) and a lower bound of 21 (Manning) have been proven.At the time of this writing (August 2004), there is no OEIS entry for the sequence m(n) yet, due to the lack of termsknown.

m(1) For n = 1, set X = 1, and C = 1. Then C does not have Property B.

m(2) For n = 2, set X = 1, 2, 3 and C = 1, 2, 1, 3, 2, 3. Then C does not have Property B, so m(2) <=3. However, C' = 1, 2, 1, 3 does (set Y = 1 and Z = 2, 3), so m(2) >= 3.

m(3) For n = 3, set X = 1, 2, 3, 4, 5, 6, 7, and C = 1, 2, 4, 2, 3, 5, 3, 4, 6, 4, 5, 7, 5, 6, 1, 6, 7, 2,7, 1, 3 (the Steiner triple system S7); C does not have Property B (so m(3) <= 7), but if any element of C isomitted, then that element can be taken as Y, and the set of remaining elements C' will have Property B (so forthis particular case, m(3) >= 7). One may check all other collections of 6 3-sets to see that all have PropertyB.

m(4) Seymour (1974) constructed a hypergraph on 11 vertices with 23 edges without Property B, which shows thatm(4) <= 23. Manning (1995) proved that m(4) >= 20.

45.2 Asymptotics of m(n)

Erdős (1963) proved that for any collection of fewer than 2n−1 sets of size n, there exists a 2-coloring in whichno set is monochromatic. The proof is simple: Consider a random coloring. The probability that an arbitrary setis monochromatic is 2−n+1 . By a union bound, the probability that there exist a monochromatic set is less than2n−12−n+1 = 1 . Therefore, there exists a good coloring.Erdős (1964) constructed an n-uniform graph with O(2n · n2) edges which does not have property B, establishingan upper bound. Schmidt (1963) proved that every collection of at most n/(n + 4) · 2n has property B. Erdős and

160

45.3. REFERENCES 161

Lovász conjectured that m(n) = θ(2n · n) . Beck in 1978 improved the lower bound to m(n) = Ω(n1/32n) . In2000, Radhakrishnan and Srinivasan improved the lower bound to m(n) = Ω(2n ·

√n/ logn) . They used a clever

probabilistic algorithm.

45.3 References• Bernstein, F. (1908), “Zur theorie der trigonometrische Reihen”, Leipz. Ber. 60: 325–328.

• Erdős, Paul (1963), “On a combinatorial problem”, Nordisk Mat. Tidskr. 11: 5–10

• Erdős, P. (1964). “On a combinatorial problem. II”. Acta Mathematica Academiae Scientiarum Hungaricae15 (3–4): 445–447. doi:10.1007/BF01897152.

• Schmidt, W. M. (1964). “Ein kombinatorisches Problem von P. Erdős und A. Hajnal”. Acta MathematicaAcademiae Scientiarum Hungaricae 15 (3–4): 373–374. doi:10.1007/BF01897145.

• Seymour, Paul (1974), “A note on a combinatorial problem of Erdős and Hajnal”, Bulletin of the LondonMathematical Society 8: 681–682, doi:10.1112/jlms/s2-8.4.681.

• Toft, Bjarne (1975), “On colour-critical hypergraphs”, in Hajnal, A.; Rado, Richard; Sós, Vera T., Infinite andFinite Sets: To Paul Erdös on His 60th Birthday, North Holland Publishing Co., pp. 1445–1457.

• Manning, G. M. (1995), “Some results on the m(4) problem of Erdős and Hajnal”, Electronic Research An-nouncements of the American Mathematical Society 1: 112–113, doi:10.1090/S1079-6762-95-03004-6.

• Beck, J. (1978), “On 3-chromatic hypergraphs”, Discrete Mathematics 24 (2): 127–137, doi:10.1016/0012-365X(78)90191-7.

• Radhakrishnan, J.; Srinivasan, A. (2000), “Improved bounds and algorithms for hypergraph 2-coloring”, Ran-dom Structures andAlgorithms 16 (1): 4–32, doi:10.1002/(SICI)1098-2418(200001)16:1<4::AID-RSA2>3.0.CO;2-2.

• Miller, E. W. (1937), “On a property of families of sets”, Comp. Rend. Varsovie: 31–38.

• Erdős, P.; Hajnal, A. (1961), “On a property of families of sets”, Acta Math. Acad. Sci. Hung. 12: 87–123,doi:10.1007/BF02066676.

• Abbott, H. L.; Hanson, D. (1969), “On a combinatorial problem of Erdös”, Canadian Mathematical Bulletin12: 823–829, doi:10.4153/CMB-1969-107-x

.

Chapter 46

Radon’s theorem

In geometry, Radon’s theorem on convex sets, named after Johann Radon, states that any set of d + 2 points in Rd

can be partitioned into two disjoint sets whose convex hulls intersect. A point in the intersection of these convex hullsis called a Radon point of the set.For example, in the case d = 2, any set of four points in the Euclidean plane can be partitioned in one of two ways. Itmay form a triple and a singleton, where the convex hull of the triple (a triangle) contains the singleton; alternatively,it may form two pairs of points that form the endpoints of two intersecting line segments.

46.1 Proof and construction

+1

+1

–1

–1

+1

+1+1

–3

Two sets of four points in the plane (the vertices of a square and an equilateral triangle with its centroid), the multipliers solving thesystem of three linear equations for these points, and the Radon partitions formed by separating the points with positive multipliersfrom the points with negative multipliers.

Consider any set x1, x2, . . . , xd+2 ⊂ Rd of d + 2 points in d-dimensional space. Then there exists a set ofmultipliers a1, ..., ad ₊ ₂, not all of which are zero, solving the system of linear equations

d+2∑i=1

aixi = 0,d+2∑i=1

ai = 0,

because there are d + 2 unknowns (the multipliers) but only d + 1 equations that they must satisfy (one for eachcoordinate of the points, together with a final equation requiring the sum of the multipliers to be zero). Fix someparticular nonzero solution a1, ..., ad ₊ ₂. Let I be the set of points with positive multipliers, and let J be the setof points with multipliers that are negative or zero. Then I and J form the required partition of the points into twosubsets with intersecting convex hulls.The convex hulls of I and J must intersect, because they both contain the point

162

46.2. TOPOLOGICAL RADON THEOREM 163

p =∑i∈I

aiAxi =

∑j∈J

−ajA

xj ,

where

A =∑i∈I

ai = −∑j∈J

aj .

The left hand side of the formula for p expresses this point as a convex combination of the points in I, and the righthand side expresses it as a convex combination of the points in J. Therefore, p belongs to both convex hulls, completingthe proof.This proof method allows for the efficient construction of a Radon point, in an amount of time that is polynomial inthe dimension, by using Gaussian elimination or other efficient algorithms to solve the system of equations for themultipliers.[1]

46.2 Topological Radon theorem

A topological generalization of Radon’s theorem states that, if ƒ is any continuous function from a (d + 1)-dimensionalsimplex to d-dimensional space, then the simplex has two disjoint faces whose images under ƒ are not disjoint.[2]

Radon’s theorem itself can be interpreted as the special case in which ƒ is the unique affine map that takes the verticesof the simplex to a given set of d + 2 points in d-dimensional space.More generally, if K is any (d + 1)-dimensional compact convex set, and ƒ is any continuous function from K tod-dimensional space, then there exists a linear function g such that some point where g achieves its maximum valueand some other point where g achieves its minimum value are mapped by ƒ to the same point. In the case where K isa simplex, the two simplex faces formed by the maximum and minimum points of g must then be two disjoint faceswhose images have a nonempty intersection. This same general statement, when applied to a hypersphere instead ofa simplex, gives the Borsuk–Ulam theorem, that ƒ must map two opposite points of the sphere to the same point.[2]

46.3 Applications

The Radon point of any four points in the plane is their geometric median, the point that minimizes the sum ofdistances to the other points.[3][4]

Radon’s theorem forms a key step of a standard proof of Helly’s theorem on intersections of convex sets;[5] this proofwas the motivation for Radon’s original discovery of Radon’s theorem.Radon’s theorem can also be used to calculate the VC dimension of d-dimensional points with respect to linearseparations. There exist sets of d + 1 points (for instance, the points of a regular simplex) such that every twononempty subsets can be separated from each other by a hyperplane. However, no matter which set of d + 2 points isgiven, the two subsets of a Radon partition cannot be linearly separated. Therefore, the VC dimension of this systemis exactly d + 1.[6]

A randomized algorithm that repeatedly replaces sets of d + 2 points by their Radon point can be used to computean approximation to a centerpoint of any point set, in an amount of time that is polynomial in both the number ofpoints and the dimension.[1]

46.4 Related concepts

The Radon point of three points in a one-dimensional space is just their median. The geometric median of a set ofpoints is the point minimizing the sum of distances to the points in the set; it generalizes the one-dimensional medianand has been studied both from the point of view of facility location and robust statistics. For sets of four points inthe plane, the geometric median coincides with the Radon point.

164 CHAPTER 46. RADON’S THEOREM

Another generalization for partition into r sets was given by Helge Tverberg (1966) and is now known as Tverberg’stheorem. It states that for any set of

(d+ 1)(r − 1) + 1

points in Euclidean d-space, there is a partition into r subsets whose convex hulls intersect in at least one commonpoint.Carathéodory’s theorem states that any point in the convex hull of some set of points is also within the convex hull ofa subset of at most d + 1 of the points; that is, that the given point is part of a Radon partition in which it is a singleton.One proof of Carathéodory’s theorem uses a technique of examining solutions to systems of linear equations, similarto the proof of Radon’s theorem, to eliminate one point at a time until at most d + 1 remain.Concepts related to Radon’s theorem have also been considered for convex geometries, families of finite sets withthe properties that the intersection of any two sets in the family remains in the family, and that the empty set and theunion of all the sets belongs to the family. In this more general context, the convex hull of a set S is the intersectionof the family members that contain S, and the Radon number of a space is the smallest r such that any r points havetwo subsets whose convex hulls intersect. Similarly, one can define the Helly number h and the Carathéodory numberc by analogy to their definitions for convex sets in Euclidean spaces, and it can be shown that these numbers satisfythe inequalities h < r ≤ ch + 1.[7]

In an arbitrary undirected graph, one may define a convex set to be a set of vertices that includes every induced pathconnecting a pair of vertices in the set. With this definition, every set of ω + 1 vertices in the graph can be partitionedinto two subsets whose convex hulls intersect, and ω + 1 is the minimum number for which this is possible, whereω is the clique number of the given graph.[8] For related results involving shortest paths instead of induced paths seeChepoi (1986) and Bandelt & Pesch (1989).

46.5 Notes[1] Clarkson et al. (1996).

[2] Bajmóczy & Bárány (1979); Matoušek (2003).

[3] Cieslik, Dietmar (2006), Shortest Connectivity: An Introduction with Applications in Phylogeny, Combinatorial Optimization17, Springer, p. 6, ISBN 9780387235394.

[4] Plastria, Frank (2006), “Four-point Fermat location problems revisited. New proofs and extensions of old results” (PDF),IMA Journal of Management Mathematics 17 (4): 387–396, doi:10.1093/imaman/dpl007, Zbl 1126.90046.

[5] Matoušek (2002), p. 11.

[6] Epsilon-nets and VC-dimension, Lecture Notes by Marco Pellegrini, 2004.

[7] Kay & Womble (1971).

[8] Duchet (1987).

46.6 References• Bajmóczy, E. G.; Bárány, I. (1979), “A common generalization of Borsuk’s and Radon’s theorem”, Acta Math-ematica Hungarica 34: 347–350, doi:10.1007/BF01896131.

• Bandelt, H.-J.; Pesch, E. (1989), “A Radon theorem for Helly graphs”, Archiv der Mathematik 52 (1): 95–98,doi:10.1007/BF01197978.

• Chepoi, V. D. (1986), “Some properties of the d-convexity in triangulated graphs”, Mat. Issled. (in Russian)87: 164–177. As cited by Bandelt & Pesch (1989).

• Clarkson, Kenneth L.; Eppstein, David; Miller, Gary L.; Sturtivant, Carl; Teng, Shang-Hua (1996), “Approximatingcenter points with iterated Radon points” (PDF), Int. J. Computational Geometry & Applications 6 (3): 357–377, doi:10.1142/s021819599600023x, MR 97h:65010.

46.6. REFERENCES 165

• Danzer, L.; Grünbaum, B.; Klee, V. (1963), “Helly’s theorem and its relatives”, Convexity, Proc. Symp. PureMath. 7, American Mathematical Society, pp. 101–179.

• Duchet, Pierre (1987), “Convex sets in graphs. II. Minimal path convexity”, Journal of Combinatorial Theory,Series A 44 (3): 307–316, doi:10.1016/0095-8956(88)90039-1. As cited by Bandelt & Pesch (1989).

• Eckhoff, J. (1993), “Helly, Radon, and Carathéodory type theorems”, Handbook of Convex Geometry A, B,Amsterdam: North-Holland, pp. 389–448.

• Kay, David C.; Womble, Eugene W. (1971), “Axiomatic convexity theory and relationships between theCarathéodory, Helly, and Radon numbers”, Pacific Journal ofMathematics 38 (2): 471–485, doi:10.2140/pjm.1971.38.471,MR 0310766.

• Matoušek, J. (2002), “1.3 Radon’s Lemma and Helly’s Theorem”, Lectures on Discrete Geometry, GraduateTexts in Mathematics 212, Springer-Verlag, pp. 9–12, ISBN 978-0-387-95373-1.

• Matoušek, J. (2003), “5.1 Nonembeddability Theorems: An Introduction”, Using the Borsuk–Ulam Theorem:Lectures on Topological Methods in Combinatorics and Geometry, Springer-Verlag, pp. 88–92.

• Radon, J. (1921), “Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten”, Mathematische An-nalen 83 (1–2): 113–115, doi:10.1007/BF01464231.

• Tverberg, H. (1966), “A generalization of Radon’s theorem” (PDF), Journal of the London MathematicalSociety 41: 123–128, doi:10.1112/jlms/s1-41.1.123.

Chapter 47

Ring of sets

Not to be confused with Ring (mathematics).

In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In ordertheory, a nonempty family of setsR is called a ring (of sets) if it is closed under intersection and union. That is, thefollowing two statements are true for all sets A and B ,

1. A,B ∈ R implies A ∩B ∈ R and

2. A,B ∈ R implies A ∪B ∈ R. [1]

In measure theory, a ring of setsR is instead a nonempty family closed under unions and set-theoretic differences.[2]

That is, the following two statements are true for all sets A and B (including when they are the same set),

1. A,B ∈ R implies A \B ∈ R and

2. A,B ∈ R implies A ∪B ∈ R.

This implies the empty set is in R . It also implies that R is closed under symmetric difference and intersection,because of the identities

1. AB = (A \B) ∪ (B \A) and

2. A ∩B = A \ (A \B).

(So a ring in the second, measure theory, sense is also a ring in the first, order theory, sense.) Together, theseoperations give R the structure of a boolean ring. Conversely, every family of sets closed under both symmetricdifference and intersection is also closed under union and differences. This is due to the identities

1. A ∪B = (AB) (A ∩B) and

2. A \B = A (A ∩B).

47.1 Examples

If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.If (X,≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongsto an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions. However,in general it will not be closed under differences of sets.The open sets and closed sets of any topological space are closed under both unions and intersections.[1]

166

47.2. RELATED STRUCTURES 167

On the real line R, the family of sets consisting of the empty set and all finite unions of intervals of the form (a, b],a,b in R is a ring in the measure theory sense.If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed underboth unions and intersections.[1]

If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form aring of sets.[1]

47.2 Related structures

A ring of sets (in the order-theoretic sense) forms a distributive lattice in which the intersection and union operationscorrespond to the lattice’s meet and join operations, respectively. Conversely, every distributive lattice is isomorphicto a ring of sets; in the case of finite distributive lattices, this is Birkhoff’s representation theorem and the sets maybe taken as the lower sets of a partially ordered set.[1]

A field of subsets of X is a ring that contains X and is closed under relative complement. Every field, and so alsoevery σ-algebra, is a ring of sets in the measure theory sense.A semi-ring (of sets) is a family of sets S with the properties

1. ∅ ∈ S,

2. A,B ∈ S implies A ∩B ∈ S, and

3. A,B ∈ S implies A \B =∪n

i=1 Ci for some disjoint C1, . . . , Cn ∈ S.

Clearly, every ring (in the measure theory sense) is a semi-ring.A semi-field of subsets of X is a semi-ring that contains X.

47.3 References[1] Birkhoff, Garrett (1937), “Rings of sets”,DukeMathematical Journal 3 (3): 443–454, doi:10.1215/S0012-7094-37-00334-

X, MR 1546000.

[2] De Barra, Gar (2003), Measure Theory and Integration, Horwood Publishing, p. 13, ISBN 9781904275046.

47.4 External links• Ring of sets at Encyclopedia of Mathematics

Chapter 48

Sauer–Shelah lemma

Pajor’s formulation of the Sauer–Shelah lemma: for every finite family of sets (green) there is another family of equally many sets(blue outlines) such that each set in the second family is shattered by the first family

In combinatorial mathematics and extremal set theory, the Sauer–Shelah lemma states that every family of setswith small VC dimension consists of a small number of sets. It is named after Norbert Sauer and Saharon Shelah,who published it independently of each other in 1972.[1][2] The same result was also published slightly earlier andagain independently, by Vladimir Vapnik and Alexey Chervonenkis, after whom the VC dimension is named.[3] Inhis paper containing the lemma, Shelah gives credit also to Micha Perles, and for this reason the lemma has also beencalled the Perles–Sauer–Shelah lemma.[4]

Buzaglo et al. call this lemma “one of the most fundamental results on VC-dimension”,[4] and it has applications inmany areas. Sauer’s motivation was in the combinatorics of set systems, while Shelah’s was in model theory and thatof Vapnik and Chervonenkis was in statistics. It has also been applied in discrete geometry[5] and graph theory.[6]

168

48.1. DEFINITIONS AND STATEMENT 169

48.1 Definitions and statement

If F = S1, S2, . . . is a family of sets, and T is another set, then T is said to be shattered by F if every subset ofT (including the empty set and T itself) can be obtained as an intersection T ∩Si between T and a set in the family.The VC dimension of F is the largest cardinality of a set shattered by F .In terms of these definitions, the Sauer–Shelah lemma states that ifF is a family of sets with n distinct elements suchthat |F| >

∑k−1i=0

(ni

), then F shatters a set of size k . Equivalently, if the VC dimension of F is k , then F can

consist of at most∑k

i=0

(ni

)= O(nk) sets.

The bound of the lemma is tight: there exists a family F with |F| =∑k−1

i=0

(ni

)that does not shatter any set of size

k . Namely, let F be the family of all subsets of 1, 2, . . . n that have cardinality less than k .[7]

48.2 The number of shattered sets

A strengthening of the Sauer–Shelah lemma, due to Pajor (1985), states that every finite set family F shatters at least|F| sets.[8] This immediately implies the Sauer–Shelah lemma, because only

∑k−1i=0

(ni

)of the subsets of an n -item

universe have cardinality less than k . Thus, when |F| >∑k−1

i=0

(ni

), there are not enough small sets to be shattered,

so one of the shattered sets must have cardinality at least k .For a restricted type of shattered set, called an order-shattered set, the number of shattered sets always equals thecardinality of the set family.[9]

48.3 Proof

Pajor’s variant of the Sauer–Shelah lemma may be proved by mathematical induction; the proof has variously beencredited to Noga Alon[10] or to Ron Aharoni and Ron Holzman.[9] As a base case to the induction, every family ofonly one set shatters the empty set. To see that every finite familyF of two or more sets shatters at least |F| sets, let xbe an element that belongs to some but not all of the sets inF . SplitF into two subfamilies, of the sets that contain xand the sets that do not contain x . By induction, these two subfamilies shatter two collections of sets whose sizes addto at least |F| . None of these shattered sets contain x , but some of them may be shattered by both subfamilies. Whena set S is shattered by only one of the two subfamilies, it contributes one unit both to the number of shattered sets ofthe subfamily and to the number of shattered sets of F . When a set S is shattered by both subfamilies, then both Sand S ∪ x are shattered by F , and S contributes two units to the number of shattered sets of the subfamilies andof F . Therefore, the number of shattered sets of F is at least equal to the number shattered by the two subfamiliesof F , which is at least |F| .A different proof of the Sauer–Shelah lemma in its original form, by Péter Frankl and János Pach, is based on linearalgebra and the inclusion–exclusion principle.[5][7]

48.4 Applications

The original application of the lemma, by Vapnik and Chervonenkis, was in showing that every probability distributioncan be approximated (with respect to a family of events of a given VC dimension) by a finite set of sample pointswhose cardinality depends only on the VC dimension of the family of events. In this context, there are two importantnotions of approximation, both parameterized by a number ε: a set S of samples, and a probability distribution on S,is said to be an ε-approximation of the original distribution if the probability of each event with respect to S differsfrom its original probability by at most ε. A set S of (unweighted) samples is said to be an ε-net if every event withprobability at least ε includes at least one point of S. An ε-approximation must also be an ε-net but not necessarilyvice versa.Vapnik and Chervonenkis used the lemma to show that set systems of VC dimension d always have ε-approximationsof cardinality O( d

ϵ2 log dϵ ) . Later authors including Haussler & Welzl (1987)[11] and Komlós, Pach & Woeginger

(1992)[12] similarly showed that there always exist ε-nets of cardinality O(dϵ log 1ϵ ) , and more precisely of cardinality

at most dϵ ln 1

ϵ +2dϵ ln ln 1

ϵ +6dϵ .[5] The main idea of the proof of the existence of small ε-nets is to choose a random

170 CHAPTER 48. SAUER–SHELAH LEMMA

sample x of cardinality O(dϵ log 1ϵ ) and a second independent random sample y of cardinality O(dϵ log2 1

ϵ ) , and tobound the probability that x is missed by some large event E by the probability that x is missed and simultaneouslythe intersection of y with E is larger than its median value. For any particular E, the probability that x is missed whiley is larger than its median is very small, and the Sauer–Shelah lemma (applied to x ∪ y ) shows that only a smallnumber of distinct events E need to be considered, so by the union bound, with nonzero probability, x is an ε-net.[5]

In turn, ε-nets and ε-approximations, and the likelihood that a random sample of large enough cardinality has theseproperties, have important applications in machine learning, in the area of probably approximately correct learning.[13]

In computational geometry, they have been applied to range searching,[11] derandomization,[14] and approximationalgorithms.[15][16]

Kozma & Moran (2013) use generalizations of the Sauer–Shelah lemma to prove results in graph theory such as thatthe number of strong orientations of a given graph is sandwiched between its numbers of connected and 2-edge-connected subgraphs.[6]

48.5 References[1] Sauer, N. (1972), “On the density of families of sets”, Journal of Combinatorial Theory, Series A 13: 145–147, doi:10.1016/0097-

3165(72)90019-2, MR 0307902.

[2] Shelah, Saharon (1972), “A combinatorial problem; stability and order for models and theories in infinitary languages”,Pacific Journal of Mathematics 41: 247–261, doi:10.2140/pjm.1972.41.247, MR 0307903.

[3] Vapnik, V. N.; Červonenkis, A. Ja. (1971), “The uniform convergence of frequencies of the appearance of events to theirprobabilities”, Akademija Nauk SSSR 16: 264–279, MR 0288823.

[4] Buzaglo, Sarit; Pinchasi, Rom; Rote, Günter (2013), “Topological hypergraphs”, in Pach, János, Thirty Essays on GeometricGraph Theory, Springer, pp. 71–81, doi:10.1007/978-1-4614-0110-0_6.

[5] Pach, János; Agarwal, Pankaj K. (1995), Combinatorial geometry, Wiley-Interscience Series in Discrete Mathematics andOptimization, New York: John Wiley & Sons Inc., p. 247, doi:10.1002/9781118033203, ISBN 0-471-58890-3, MR1354145.

[6] Kozma, László; Moran, Shay (2013), “Shattering, Graph Orientations, and Connectivity”, Electronic Journal of Combina-torics 20 (3), P44, arXiv:1211.1319.

[7] Gowers, Timothy (July 31, 2008), “Dimension arguments in combinatorics”, Gowers’s Weblog: Mathematics related dis-cussions, Example 3.

[8] Pajor, Alain (1985), Sous-espaces ln1 des espaces de Banach, Travaux en Cours [Works in Progress] 16, Paris: Hermann,ISBN 2-7056-6021-6, MR 903247. As cited by Anstee, Rónyai & Sali (2002).

[9] Anstee, R. P.; Rónyai, Lajos; Sali, Attila (2002), “Shattering news”,Graphs and Combinatorics 18 (1): 59–73, doi:10.1007/s003730200003,MR 1892434.

[10] Kalai, Gil (September 28, 2008), “Extremal Combinatorics III: Some Basic Theorems”, Combinatorics and More.

[11] Haussler, David; Welzl, Emo (1987), "ε-nets and simplex range queries”, Discrete and Computational Geometry 2 (2):127–151, doi:10.1007/BF02187876, MR 884223.

[12] Komlós, János; Pach, János; Woeginger, Gerhard (1992), “Almost tight bounds for ε-nets”, Discrete and ComputationalGeometry 7 (2): 163–173, doi:10.1007/BF02187833, MR 1139078.

[13] Blumer, Anselm; Ehrenfeucht, Andrzej; Haussler, David; Warmuth, Manfred K. (1989), “Learnability and the Vapnik–Chervonenkis dimension”, Journal of the ACM 36 (4): 929–965, doi:10.1145/76359.76371, MR 1072253.

[14] Chazelle, B.; Friedman, J. (1990), “A deterministic view of random sampling and its use in geometry”, Combinatorica 10(3): 229–249, doi:10.1007/BF02122778, MR 1092541.

[15] Brönnimann, H.; Goodrich, M. T. (1995), “Almost optimal set covers in finite VC-dimension”, Discrete and ComputationalGeometry 14 (4): 463–479, doi:10.1007/BF02570718, MR 1360948.

[16] Har-Peled, Sariel (2011), “On complexity, sampling, and ε-nets and ε-samples”, Geometric approximation algorithms,Mathematical Surveys and Monographs 173, Providence, RI: American Mathematical Society, pp. 61–85, ISBN 978-0-8218-4911-8, MR 2760023.

Chapter 49

Segal space

In mathematics, a Segal space is a simplicial space satisfying some extra conditions. Complete Segal spaces wereintroduced by Rezk (2001) as models for (∞, 1)-categories.

49.1 References• Rezk, Charles (2001), “A model for the homotopy theory of homotopy theory”, Transactions of the Ameri-can Mathematical Society 353 (3): 973–1007, doi:10.1090/S0002-9947-00-02653-2, ISSN 0002-9947, MR1804411

49.2 External links• Segal space in nLab

• Complete Segal space in nLab

171

Chapter 50

Set cover problem

The set cover problem is a classical question in combinatorics, computer science and complexity theory. It is one ofKarp’s 21 NP-complete problems shown to be NP-complete in 1972.It is a problem “whose study has led to the development of fundamental techniques for the entire field” of approximationalgorithms.[1]

Given a set of elements 1, 2, ...,m (called the universe) and a set S of n sets whose union equals the universe, theset cover problem is to identify the smallest subset of S whose union equals the universe. For example, consider theuniverse U = 1, 2, 3, 4, 5 and the set of sets S = 1, 2, 3, 2, 4, 3, 4, 4, 5 . Clearly the union of S is U. However, we can cover all of the elements with the following, smaller number of sets: 1, 2, 3, 4, 5 .More formally, given a universe U and a family S of subsets of U , a cover is a subfamily C ⊆ S of sets whose unionis U . In the set covering decision problem, the input is a pair (U ,S) and an integer k ; the question is whether thereis a set covering of size k or less. In the set covering optimization problem, the input is a pair (U ,S) , and the taskis to find a set covering that uses the fewest sets.The decision version of set covering is NP-complete, and the optimization version of set cover is NP-hard .[2]

If each set is assigned a cost, it becomes a weighted set cover problem.

50.1 Integer linear program formulation

The minimum set cover problem can be formulated as the following integer linear program (ILP).[3]

This ILP belongs to the more general class of ILPs for covering problems. The integrality gap of this ILP is at mostlog n , so its relaxation gives a factor- log n approximation algorithm for the minimum set cover problem (where n isthe size of the universe).[4]

50.2 Hitting set formulation

Set covering is equivalent to the hitting set problem. It is easy to see this by observing that an instance of set coveringcan be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, the universe representedby vertices on the right, and edges representing the inclusion of elements in sets. The task is then to find a minimumcardinality subset of left-vertices which covers all of the right-vertices. In the Hitting set problem, the objective isto cover the left-vertices using a minimum subset of the right vertices. Converting from one problem to the other istherefore achieved by interchanging the two sets of vertices.

50.3 Greedy algorithm

There is a greedy algorithm for polynomial time approximation of set covering that chooses sets according to onerule: at each stage, choose the set that contains the largest number of uncovered elements. It can be shown[5] that this

172

50.4. LOW-FREQUENCY SYSTEMS 173

algorithm achieves an approximation ratio of H(s) , where s is the size of the set to be covered, H(n) is the n -thharmonic number:

H(n) =n∑

k=1

1

k≤ lnn+ 1

This greedy algorithm actually achieves an approximation ratio of H(s′) where s′ is the maximum cardinality set ofS . For δ-dense instances, there exists, however, a c lnm -approximation algorithm for every c > 0 .[6]

Tight example for the greedy algorithm with k=3

There is a standard example on which the greedy algorithm achieves an approximation ratio of log2(n)/2 . Theuniverse consists of n = 2(k+1) − 2 elements. The set system consists of k pairwise disjoint sets S1, . . . , Sk withsizes 2, 4, 8, . . . , 2k respectively, as well as two additional disjoint sets T0, T1 , each of which contains half of theelements from each Si . On this input, the greedy algorithm takes the sets Sk, . . . , S1 , in that order, while the optimalsolution consists only of T0 and T1 . An example of such an input for k = 3 is pictured on the right.Inapproximability results show that the greedy algorithm is essentially the best-possible polynomial time approxima-tion algorithm for set cover (see Inapproximability results below), under plausible complexity assumptions.

50.4 Low-frequency systems

If each element occurs in at most f sets, then a solution can be found in polynomial time that approximates theoptimum to within a factor of f using LP relaxation.[7]

50.5 Inapproximability results

Whenn refers to the size of the universe, Lund & Yannakakis (1994) showed that set covering cannot be approximatedin polynomial time to within a factor of 1

2 log2 n ≈ 0.72 lnn , unlessNP has quasi-polynomial time algorithms. Feige(1998) improved this lower bound to

(1 − o(1)

)· lnn under the same assumptions, which essentially matches the

approximation ratio achieved by the greedy algorithm. Raz & Safra (1997) established a lower bound of c · lnn ,where c is a certain constant, under the weaker assumption that P = NP. A similar result with a higher value of c wasrecently proved by Alon, Moshkovitz & Safra (2006). Dinur & Steurer (2013) showed optimal inapproximability byproving that it cannot be approximated to

(1− o(1)

)· lnn unless P = NP.

50.6 Related problems• Hitting set is an equivalent reformulation of Set Cover.

• Vertex cover is a special case of Hitting Set.

174 CHAPTER 50. SET COVER PROBLEM

• Edge cover is a special case of Set Cover.

• Set packing is the dual problem of Set Cover.

• Maximum coverage problem is to choose at most k sets to cover as many elements as possible.

• Dominating set is the problem of selecting a set of vertices (the dominating set) in a graph such that all othervertices are adjacent to at least one vertex in the dominating set. The Dominating set problem was shown tobe NP complete through a reduction from Set cover.

• Exact cover problem is to choose a set cover with no element included in more than one covering set.

• Closest pair of points problem

• Nearest neighbor search

50.7 Notes[1] Vazirani (2001, p. 15)

[2] Korte & Vygen 2012, p. 414.

[3] Vazirani (2001, p. 108)

[4] Vazirani (2001, pp. 110–112)

[5] Chvatal, V. A Greedy Heuristic for the Set-Covering Problem. Mathematics of Operations Research Vol. 4, No. 3 (Aug.,1979), pp. 233-235

[6] Karpinski & Zelikovsky 1998

[7] Vazirani (2001, pp. 118–119)

50.8 References• Alon, Noga; Moshkovitz, Dana; Safra, Shmuel (2006), “Algorithmic construction of sets for k-restrictions”,ACM Trans. Algorithms (ACM) 2 (2): 153–177, doi:10.1145/1150334.1150336, ISSN 1549-6325.

• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), Introduction to Algo-rithms, Cambridge, Mass.: MIT Press and McGraw-Hill, pp. 1033–1038, ISBN 0-262-03293-7

• Feige, Uriel (1998), “A threshold of ln n for approximating set cover”, Journal of the ACM (ACM) 45 (4):634–652, doi:10.1145/285055.285059, ISSN 0004-5411.

• Karpinski, Marek; Zelikovsky, Alexander (1998). “Approximating dense cases of covering problems”. Pro-ceedings of the DIMACS Workshop on Network Design: Connectivity and Facilities Location 40. pp. 169–178.

• Lund, Carsten; Yannakakis, Mihalis (1994), “On the hardness of approximating minimization problems”,Journal of the ACM (ACM) 41 (5): 960–981, doi:10.1145/185675.306789, ISSN 0004-5411.

• Raz, Ran; Safra, Shmuel (1997), “A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP”, STOC '97: Proceedings of the twenty-ninth annual ACM symposiumon Theory of computing, ACM, pp. 475–484, ISBN 978-0-89791-888-6.

• Dinur, Irit; Steurer, David (1997), “Analytical approach to parallel repetition”, STOC '14: Proceedings of theforty-sixth annual ACM symposium on Theory of computing, ACM, pp. 624–633.

• Vazirani, Vijay V. (2001), Approximation Algorithms (PDF), Springer-Verlag, ISBN 3-540-65367-8

• Korte, Bernhard; Vygen, Jens (2012), Combinatorial Optimization: Theory and Algorithms (5 ed.), Springer,ISBN 978-3-642-24487-2

• Cardoso, Nuno; Abreu, Rui (2014). “An Efficient Distributed Algorithm for Computing Minimal Hitting Sets”(PDF). Proceedings of the 25th International Workshop on Principles of Diagnosis (DX'14, best paper award).Graz, Austria. doi:10.5281/zenodo.10037.

50.9. EXTERNAL LINKS 175

50.9 External links• Benchmarks with Hidden Optimum Solutions for Set Covering, Set Packing and Winner Determination

• A compendium of NP optimization problems - Minimum Set Cover

Chapter 51

Shapley–Folkman lemma

The Shapley–Folkman lemma is illustrated by the Minkowski addition of four sets. The point (+) in the convex hull of the Minkowskisum of the four non-convex sets (right) is the sum of four points (+) from the (left-hand) sets—two points in two non-convex setsplus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points(shown as red dots).[1]

The Shapley–Folkman lemma is a result in convex geometry with applications in mathematical economics thatdescribes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets’members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero,one, and two:

0, 1 + 0, 1 = 0 + 0, 0 + 1, 1 + 0, 1 + 1 = 0, 1, 2.

The Shapley–Folkman lemma and related results provide an affirmative answer to the question, “Is the sum of manysets close to being convex?"[2] A set is defined to be convex if every line segment joining two of its points is a subsetin the set: For example, the solid disk • is a convex set but the circle is not, because the line segment joining twodistinct points ⊘ is not a subset of the circle. The Shapley–Folkman lemma suggests that if the number of summedsets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex.[1]

The Shapley–Folkman lemma was introduced as a step in the proof of the Shapley–Folkman theorem, which statesan upper bound on the distance between the Minkowski sum and its convex hull. The convex hull of a set Q is thesmallest convex set that contains Q. This distance is zero if and only if the sum is convex. The theorem’s bound on thedistance depends on the dimensionD and on the shapes of the summand-sets, but not on the number of summand-setsN, when N > D. The shapes of a subcollection of only D summand-sets determine the bound on the distance betweenthe Minkowski average of N sets

176

51.1. INTRODUCTORY EXAMPLE 177

1⁄N (Q1 + Q2 + ... + QN)

and its convex hull. As N increases to infinity, the bound decreases to zero (for summand-sets of uniformly boundedsize).[3] The Shapley–Folkman theorem’s upper bound was decreased byStarr’s corollary (alternatively, theShapley–Folkman–Starr theorem).The lemma of Lloyd Shapley and Jon Folkman was first published by the economist Ross M. Starr, who was in-vestigating the existence of economic equilibria while studying with Kenneth Arrow.[1] In his paper, Starr studied aconvexified economy, in which non-convex sets were replaced by their convex hulls; Starr proved that the convexifiedeconomy has equilibria that are closely approximated by “quasi-equilibria” of the original economy; moreover, heproved that every quasi-equilibrium has many of the optimal properties of true equilibria, which are proved to exist forconvex economies. Following Starr’s 1969 paper, the Shapley–Folkman–Starr results have been widely used to showthat central results of (convex) economic theory are good approximations to large economies with non-convexities; forexample, quasi-equilibria closely approximate equilibria of a convexified economy. “The derivation of these resultsin general form has been one of the major achievements of postwar economic theory”, wrote Roger Guesnerie.[4]

The topic of non-convex sets in economics has been studied by many Nobel laureates, besides Lloyd Shapley whowon the prize in 2012: Arrow (1972), Robert Aumann (2005), Gérard Debreu (1983), Tjalling Koopmans (1975),Paul Krugman (2008), and Paul Samuelson (1970); the complementary topic of convex sets in economics has beenemphasized by these laureates, along with Leonid Hurwicz, Leonid Kantorovich (1975), and Robert Solow (1987).The Shapley–Folkman lemma has applications also in optimization and probability theory.[3] In optimization theory,the Shapley–Folkman lemma has been used to explain the successful solution of minimization problems that aresums of many functions.[5][6] The Shapley–Folkman lemma has also been used in proofs of the “law of averages” forrandom sets, a theorem that had been proved for only convex sets.[7]

51.1 Introductory example

For example, the subset of the integers 0, 1, 2 is contained in the interval of real numbers [0, 2], which is convex.The Shapley–Folkman lemma implies that every point in [0, 2] is the sum of an integer from 0, 1 and a real numberfrom [0, 1].[8]

The distance between the convex interval [0, 2] and the non-convex set 0, 1, 2 equals one-half

1/2 = |1 − 1/2| = |0 − 1/2| = |2 − 3/2| = |1 − 3/2|.

However, the distance between the average Minkowski sum

1/2 ( 0, 1 + 0, 1 ) = 0, 1/2, 1

and its convex hull [0, 1] is only 1/4, which is half the distance (1/2) between its summand 0, 1 and [0, 1]. Asmore sets are added together, the average of their sum “fills out” its convex hull: The maximum distance between theaverage and its convex hull approaches zero as the average includes more summands.[8]

51.2 Preliminaries

The Shapley–Folkman lemma depends upon the following definitions and results from convex geometry.

51.2.1 Real vector spaces

A real vector space of two dimensions can be given a Cartesian coordinate system in which every point is identifiedby an ordered pair of real numbers, called “coordinates”, which are conventionally denoted by x and y. Two pointsin the Cartesian plane can be added coordinate-wise

(x1, y1) + (x2, y2) = (x1+x2, y1+y2);

178 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

further, a point can be multiplied by each real number λ coordinate-wise

λ (x, y) = (λx, λy).

More generally, any real vector space of (finite) dimension D can be viewed as the set of all D-tuples of D realnumbers (v1, v2, . . . , vD) on which two operations are defined: vector addition and multiplication by a realnumber. For finite-dimensional vector spaces, the operations of vector addition and real-number multiplication caneach be defined coordinate-wise, following the example of the Cartesian plane.[9]

51.2.2 Convex sets

In a convex set Q, the line segment connecting any two of its points is a subset of Q.

In a non-convex set Q, a point in some line-segment joining two of its points is not a member of Q.Line segments test whether a subset be convex.

In a real vector space, a non-empty set Q is defined to be convex if, for each pair of its points, every point on the linesegment that joins them is a subset of Q. For example, a solid disk • is convex but a circle is not, because it doesnot contain a line segment joining its points ⊘ ; the non-convex set of three integers 0, 1, 2 is contained in theinterval [0, 2], which is convex. For example, a solid cube is convex; however, anything that is hollow or dented, forexample, a crescent shape, is non-convex. The empty set is convex, either by definition[10] or vacuously, dependingon the author.More formally, a set Q is convex if, for all points v0 and v1 in Q and for every real number λ in the unit interval [0,1],the point

(1 − λ) v0 + λv1

is a member of Q.By mathematical induction, a set Q is convex if and only if every convex combination of members of Q also belongsto Q. By definition, a convex combination of an indexed subset v0, v1, . . . , vD of a vector space is any weightedaverage λ0v0 + λ1v1 + . . . + λDvD, for some indexed set of non-negative real numbers λ satisfying the equationλ0 + λ1 + . . . + λD = 1.[11]

The definition of a convex set implies that the intersection of two convex sets is a convex set. More generally, theintersection of a family of convex sets is a convex set. In particular, the intersection of two disjoint sets is the emptyset, which is convex.[10]

51.2. PRELIMINARIES 179

In the convex hull of the red set, each blue point is a convex combination of some red points.

51.2.3 Convex hull

For every subset Q of a real vector space, its convex hull Conv(Q) is the minimal convex set that contains Q. ThusConv(Q) is the intersection of all the convex sets that cover Q. The convex hull of a set can be equivalently defined tobe the set of all convex combinations of points in Q.[12] For example, the convex hull of the set of integers 0,1 isthe closed interval of real numbers [0,1], which contains the integer end-points.[8] The convex hull of the unit circleis the closed unit disk, which contains the unit circle.

51.2.4 Minkowski addition

In a real vector space, the Minkowski sum of two (non-empty) sets Q1 and Q2 is defined to be the set Q1 + Q2 formedby the addition of vectors element-wise from the summand sets

Q1 + Q2 = q1 + q2 : q1 ∈ Q1 and q2 ∈ Q2 .[13]

For example

0, 1 + 0, 1 = 0+0, 0+1, 1+0, 1+1 = 0, 1, 2.[8]

By the principle of mathematical induction, the Minkowski sum of a finite family of (non-empty) sets

180 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

Minkowski addition of sets. The sum of the squares Q1=[0,1]2 and Q2=[1,2]2 is the square Q1+Q2=[1,3]2.

Q : Q ≠ Ø and 1 ≤ n ≤ N

is the set formed by element-wise addition of vectors

∑ Q = ∑ q : q ∈ Q .[14]

51.2.5 Convex hulls of Minkowski sums

Minkowski addition behaves well with respect to "convexification"—the operation of taking convex hulls. Specifically,for all subsets Q1 and Q2 of a real vector space, the convex hull of their Minkowski sum is the Minkowski sum oftheir convex hulls. That is,

Conv( Q1 + Q2 ) = Conv( Q1 ) + Conv( Q2 ).

This result holds more generally, as a consequence of the principle of mathematical induction. For each finite col-lection of sets,

Conv( ∑ Q ) = ∑ Conv( Q ).[15][16]

51.3. STATEMENTS 181

Minkowski addition and convex hulls. The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convexsets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The rightplus-sign is the sum of the left plus-signs.

51.3 Statements

The preceding identity Conv( ∑ Q ) = ∑ Conv( Q ) implies that if a point x lies in the convex hull of the Minkowskisum of N sets

x ∈ Conv( ∑ Q )

then x lies in the sum of the convex hulls of the summand-sets

x ∈ ∑ Conv( Q ).

By the definition of Minkowski addition, this last expression means that x = ∑ q for some selection of points q inthe convex hulls of the summand-sets, that is, where each q ∈ Conv(Q ). In this representation, the selection of thesummand-points q depends on the chosen sum-point x.

51.3.1 Lemma of Shapley and Folkman

For this representation of the point x, the Shapley–Folkman lemma states that if the dimension D is less than thenumber of summands

D < N

then convexification is needed for only D summand-sets, whose choice depends on x: The point has a representation

x =∑

1≤d≤D

qd +∑

D+1≤n≤N

qn

where q belongs to the convex hull of Q for D (or fewer) summand-sets and q belongs to Q itself for the remainingsets. That is,

x ∈∑

1≤d≤D

Conv (Qd) +∑

D+1≤n≤N

Qn

182 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

A Winner of the 2012 Nobel Award in Economics, Lloyd Shapley proved the Shapley–Folkman lemma with Jon Folkman.[1]

for some re-indexing of the summand sets; this re-indexing depends on the particular point x being represented.[17]

The Shapley–Folkman lemma implies, for example, that every point in [0, 2] is the sum of an integer from 0, 1and a real number from [0, 1].[8]

51.3. STATEMENTS 183

Dimension of a real vector space

Conversely, the Shapley–Folkman lemma characterizes the dimension of finite-dimensional, real vector spaces. Thatis, if a vector space obeys the Shapley–Folkman lemma for a natural number D, and for no number less than D, thenits dimension is exactly D;[18] the Shapley–Folkman lemma holds for only finite-dimensional vector spaces.[19]

51.3.2 Shapley–Folkman theorem and Starr’s corollary

The circumradius (blue) and inner radius (green) of a point set (dark red, with its convex hull shown as the lighter red dashed lines).The inner radius is smaller than the circumradius except for subsets of a single circle, for which they are equal.

Shapley and Folkman used their lemma to prove their theorem, which bounds the distance between a Minkowski sumand its convex hull, the "convexified" sum:

• The Shapley–Folkman theorem states that the squared Euclidean distance from any point in the convexified sumConv( ∑ Qn ) to the original (unconvexified) sum ∑ Qn is bounded by the sum of the squares of the D largestcircumradii of the sets Qn (the radii of the smallest spheres enclosing these sets).[20] This bound is independentof the number of summand-sets N (if N > D).[21]

The Shapley–Folkman theorem states a bound on the distance between the Minkowski sum and its convex hull; thisdistance is zero if and only if the sum is convex. Their bound on the distance depends on the dimension D and on theshapes of the summand-sets, but not on the number of summand-sets N, when N > D.[3]

184 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

The circumradius often exceeds (and cannot be less than) the inner radius:[22]

• The inner radius of a set Qn is defined to be the smallest number r such that, for any point q in the convex hullof Qn, there is a sphere of radius r that contains a subset of Qn whose convex hull contains q.

Starr used the inner radius to reduce the upper bound stated in the Shapley–Folkman theorem:

• Starr’s corollary to the Shapley–Folkman theorem states that the squared Euclidean distance from any point xin the convexified sum Conv( ∑ Qn ) to the original (unconvexified) sum ∑ Qn is bounded by the sum of thesquares of the D largest inner-radii of the sets Qn.[22][23]

Starr’s corollary states an upper bound on the Euclidean distance between the Minkowski sum ofN sets and the convexhull of the Minkowski sum; this distance between the sum and its convex hull is a measurement of the non-convexityof the set. For simplicity, this distance is called the "non-convexity" of the set (with respect to Starr’s measurement).Thus, Starr’s bound on the non-convexity of the sum depends on only the D largest inner radii of the summand-sets;however, Starr’s bound does not depend on the number of summand-sets N, when N > D. For example, the distancebetween the convex interval [0, 2] and the non-convex set 0, 1, 2 equals one-half

1/2 = |1 − 1/2| = |0 − 1/2| = |2 − 3/2| = |1 − 3/2|.

Thus, Starr’s bound on the non-convexity of the average

1⁄N ∑ Qn

decreases as the number of summands N increases. For example, the distance between the averaged set

1/2 ( 0, 1 + 0, 1 ) = 0, 1/2, 1

and its convex hull [0, 1] is only 1/4, which is half the distance (1/2) between its summand 0, 1 and [0, 1]. Theshapes of a subcollection of only D summand-sets determine the bound on the distance between the average set andits convex hull; thus, as the number of summands increases to infinity, the bound decreases to zero (for summand-setsof uniformly bounded size).[3] In fact, Starr’s bound on the non-convexity of this average set decreases to zero as thenumber of summands N increases to infinity (when the inner radii of all the summands are bounded by the samenumber).[3]

51.3.3 Proofs and computations

The original proof of the Shapley–Folkman lemma established only the existence of the representation, but did notprovide an algorithm for computing the representation: Similar proofs have been given by Arrow and Hahn,[24]

Cassels,[25] and Schneider,[26] among others. An abstract and elegant proof by Ekeland has been extended byArtstein.[27][28] Different proofs have appeared in unpublished papers, also.[2][29] In 1981, Starr published an iterativemethod for computing a representation of a given sum-point; however, his computational proof provides a weakerbound than does the original result.[30] An elementary proof of the Shapley–Folkman lemma in finite-dimensionalspace can be found in the book by Bertsekas[31] together with applications in estimating the duality gap in separableoptimization problems and zero-sum games.

51.4 Applications

The Shapley–Folkman lemma enables researchers to extend results for Minkowski sums of convex sets to sums ofgeneral sets, which need not be convex. Such sums of sets arise in economics, in mathematical optimization, and inprobability theory; in each of these three mathematical sciences, non-convexity is an important feature of applications.

51.4. APPLICATIONS 185

Good

Y

Good X

I 3I 2I 1

Qx

Qy

The consumer prefers every basket of goods on the indifference curve I3 over each basket on I2. The basket (Qx, Qy), where thebudget line (shown in blue) supports I2, is optimal and also feasible, unlike any basket lying on I3 which is preferred but unfeasible.

51.4.1 Economics

See also: Convexity in economics

In economics, a consumer’s preferences are defined over all “baskets” of goods. Each basket is represented as anon-negative vector, whose coordinates represent the quantities of the goods. On this set of baskets, an indifferencecurve is defined for each consumer; a consumer’s indifference curve contains all the baskets of commodities that theconsumer regards as equivalent: That is, for every pair of baskets on the same indifference curve, the consumer doesnot prefer one basket over another. Through each basket of commodities passes one indifference curve. A consumer’spreference set (relative to an indifference curve) is the union of the indifference curve and all the commodity basketsthat the consumer prefers over the indifference curve. A consumer’s preferences are convex if all such preference setsare convex.[32]

An optimal basket of goods occurs where the budget-line supports a consumer’s preference set, as shown in thediagram. This means that an optimal basket is on the highest possible indifference curve given the budget-line, whichis defined in terms of a price vector and the consumer’s income (endowment vector). Thus, the set of optimal basketsis a function of the prices, and this function is called the consumer’s demand. If the preference set is convex, thenat every price the consumer’s demand is a convex set, for example, a unique optimal basket or a line-segment ofbaskets.[33]

186 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

Non-convex preferences

When the consumer’s preferences have concavities, the consumer may jump between two separate optimal baskets.

See also: Non-convexity (economics)

However, if a preference set is non-convex, then some prices determine a budget-line that supports two separateoptimal-baskets. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that azoo’s budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equallyvaluable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeperdoes not want to purchase half of an eagle and half of a lion (or a griffin)! Thus, the zoo-keeper’s preferences arenon-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.[34]

When the consumer’s preference set is non-convex, then (for some prices) the consumer’s demand is not connected;a disconnected demand implies some discontinuous behavior by the consumer, as discussed by Harold Hotelling:

If indifference curves for purchases be thought of as possessing a wavy character, convex to theorigin in some regions and concave in others, we are forced to the conclusion that it is only the portions

51.4. APPLICATIONS 187

convex to the origin that can be regarded as possessing any importance, since the others are essentiallyunobservable. They can be detected only by the discontinuities that may occur in demand with variationin price-ratios, leading to an abrupt jumping of a point of tangency across a chasm when the straight lineis rotated. But, while such discontinuities may reveal the existence of chasms, they can never measuretheir depth. The concave portions of the indifference curves and their many-dimensional generalizations,if they exist, must forever remain in unmeasurable obscurity.[35]

The difficulties of studying non-convex preferences were emphasized by Herman Wold[36] and again by Paul Samuel-son, who wrote that non-convexities are “shrouded in eternal darkness ...”,[37] according to Diewert.[38]

Nonetheless, non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in The Journalof Political Economy (JPE). The main contributors were Farrell,[39] Bator,[40] Koopmans,[41] and Rothenberg.[42]

In particular, Rothenberg’s paper discussed the approximate convexity of sums of non-convex sets.[43] These JPE-papers stimulated a paper by Lloyd Shapley and Martin Shubik, which considered convexified consumer-preferencesand introduced the concept of an “approximate equilibrium”.[44] The JPE-papers and the Shapley–Shubik paperinfluenced another notion of “quasi-equilibria”, due to Robert Aumann.[45][46]

Starr’s 1969 paper and contemporary economics

Previous publications on non-convexity and economics were collected in an annotated bibliography by Kenneth Ar-row. He gave the bibliography to Starr, who was then an undergraduate enrolled in Arrow’s (graduate) advancedmathematical-economics course.[47] In his term-paper, Starr studied the general equilibria of an artificial economyin which non-convex preferences were replaced by their convex hulls. In the convexified economy, at each price, theaggregate demand was the sum of convex hulls of the consumers’ demands. Starr’s ideas interested the mathemati-cians Lloyd Shapley and Jon Folkman, who proved their eponymous lemma and theorem in “private correspondence”,which was reported by Starr’s published paper of 1969.[1]

In his 1969 publication, Starr applied the Shapley–Folkman–Starr theorem. Starr proved that the “convexified”economy has general equilibria that can be closely approximated by "quasi-equilibria" of the original economy, whenthe number of agents exceeds the dimension of the goods: Concretely, Starr proved that there exists at least onequasi-equilibrium of prices pₒ with the following properties:

• For each quasi-equilibrium’s prices pₒ , all consumers can choose optimal baskets (maximally preferred andmeeting their budget constraints).

• At quasi-equilibrium prices pₒ in the convexified economy, every good’s market is in equilibrium: Its supplyequals its demand.

• For each quasi-equilibrium, the prices “nearly clear” the markets for the original economy: an upper bound onthe distance between the set of equilibria of the “convexified” economy and the set of quasi-equilibria of theoriginal economy followed from Starr’s corollary to the Shapley–Folkman theorem.[48]

Starr established that

“in the aggregate, the discrepancy between an allocation in the fictitious economy generated by [tak-ing the convex hulls of all of the consumption and production sets] and some allocation in the realeconomy is bounded in a way that is independent of the number of economic agents. Therefore, theaverage agent experiences a deviation from intended actions that vanishes in significance as the numberof agents goes to infinity”.[49]

Following Starr’s 1969 paper, the Shapley–Folkman–Starr results have been widely used in economic theory. RogerGuesnerie summarized their economic implications: “Some key results obtained under the convexity assumptionremain (approximately) relevant in circumstances where convexity fails. For example, in economies with a largeconsumption side, preference nonconvexities do not destroy the standard results”.[50] “The derivation of these resultsin general form has been one of the major achievements of postwar economic theory”, wrote Guesnerie.[4] The topicof non-convex sets in economics has been studied by many Nobel laureates: Arrow (1972), Robert Aumann (2005),Gérard Debreu (1983), Tjalling Koopmans (1975), Paul Krugman (2008), and Paul Samuelson (1970); the com-plementary topic of convex sets in economics has been emphasized by these laureates, along with Leonid Hurwicz,

188 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

Kenneth Arrow (1972 Nobel laureate) helped Ross M. Starr to study non-convex economies.[47]

Leonid Kantorovich (1975), and Robert Solow (1987).[51] The Shapley–Folkman–Starr results have been featuredin the economics literature: in microeconomics,[52] in general-equilibrium theory,[53][54] in public economics[55] (in-cluding market failures),[56] as well as in game theory,[57] in mathematical economics,[58] and in applied mathematics(for economists).[59][60] The Shapley–Folkman–Starr results have also influenced economics research using measureand integration theory.[61]

51.4. APPLICATIONS 189

A function is convex if the region above its graph is a convex set.

51.4.2 Mathematical optimization

The Shapley–Folkman lemma has been used to explain why large minimization problems with non-convexities can benearly solved (with iterative methods whose convergence proofs are stated for only convex problems). The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of manyfunctions.[62]

Preliminaries of optimization theory

Nonlinear optimization relies on the following definitions for functions:

• The graph of a function f is the set of the pairs of arguments x and function evaluations f(x)

Graph(f) = (x, f(x) )

• The epigraph of a real-valued function f is the set of points above the graph

The sine function is non-convex.

190 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

Epi(f) = (x, u) : f(x) ≤ u .

• A real-valued function is defined to be a convex function if its epigraph is a convex set.[63]

For example, the quadratic function f(x) = x2 is convex, as is the absolute value function g(x) = |x|. However, the sinefunction (pictured) is non-convex on the interval (0, π).

Additive optimization problems

In many optimization problems, the objective function f is separable: that is, f is the sum ofmany summand-functions,each of which has its own argument:

f(x) = f( (x1, ..., xN) ) = ∑ fn(xn).

For example, problems of linear optimization are separable. Given a separable problem with an optimal solution, wefix an optimal solution

x ᵢ = (x1, ..., xN) ᵢ

with the minimum value f(x ᵢ ). For this separable problem, we also consider an optimal solution (x ᵢ , f(x ᵢ ) ) tothe "convexified problem", where convex hulls are taken of the graphs of the summand functions. Such an optimalsolution is the limit of a sequence of points in the convexified problem

(xj, f(x ) ) ∈ ∑ Conv (Graph( fn ) ).[5][64]

Of course, the given optimal-point is a sum of points in the graphs of the original summands and of a small numberof convexified summands, by the Shapley–Folkman lemma.This analysis was published by Ivar Ekeland in 1974 to explain the apparent convexity of separable problems withmany summands, despite the non-convexity of the summand problems. In 1973, the young mathematician ClaudeLemaréchal was surprised by his success with convex minimization methods on problems that were known to benon-convex; for minimizing nonlinear problems, a solution of the dual problem problem need not provide usefulinformation for solving the primal problem, unless the primal problem be convex and satisfy a constraint qualifi-cation. Lemaréchal’s problem was additively separable, and each summand function was non-convex; nonetheless,a solution to the dual problem provided a close approximation to the primal problem’s optimal value.[65][5][66] Eke-land’s analysis explained the success of methods of convex minimization on large and separable problems, despite thenon-convexities of the summand functions. Ekeland and later authors argued that additive separability produced anapproximately convex aggregate problem, even though the summand functions were non-convex. The crucial step inthese publications is the use of the Shapley–Folkman lemma.[5][66][67] The Shapley–Folkman lemma has encouragedthe use of methods of convex minimization on other applications with sums of many functions.[5][6][59][62]

51.4.3 Probability and measure theory

Convex sets are often studied with probability theory. Each point in the convex hull of a (non-empty) subset Q of afinite-dimensional space is the expected value of a simple random vector that takes its values in Q, as a consequenceof Carathéodory’s lemma. Thus, for a non-empty set Q, the collection of the expected values of the simple, Q-valued random vectors equalsQ ' s convex hull; this equality implies that the Shapley–Folkman–Starr results are usefulin probability theory.[68] In the other direction, probability theory provides tools to examine convex sets generallyand the Shapley–Folkman–Starr results specifically.[69] The Shapley–Folkman–Starr results have been widely usedin the probabilistic theory of random sets,[70] for example, to prove a law of large numbers,[7][71] a central limittheorem,[71][72] and a large-deviations principle.[73] These proofs of probabilistic limit theorems used the Shapley–Folkman–Starr results to avoid the assumption that all the random sets be convex.A probability measure is a finite measure, and the Shapley–Folkman lemma has applications in non-probabilisticmeasure theory, such as the theories of volume and of vector measures. The Shapley–Folkman lemma enables arefinement of the Brunn–Minkowski inequality, which bounds the volume of sums in terms of the volumes of their

51.5. NOTES 191

summand-sets.[74] The volume of a set is defined in terms of the Lebesgue measure, which is defined on subsetsof Euclidean space. In advanced measure-theory, the Shapley–Folkman lemma has been used to prove Lyapunov’stheorem, which states that the range of a vector measure is convex.[75] Here, the traditional term "range" (alternatively,“image”) is the set of values produced by the function. A vector measure is a vector-valued generalization of a measure;for example, if p1 and p2 are probability measures defined on the same measurable space, then the product functionp1 p2 is a vector measure, where p1 p2 is defined for every event ω by

(p1 p2)(ω)=(p1(ω), p2(ω)).

Lyapunov’s theorem has been used in economics,[45][76] in (“bang-bang”) control theory, and in statistical theory.[77]

Lyapunov’s theorem has been called a continuous counterpart of the Shapley–Folkman lemma,[3] which has itselfbeen called a discrete analogue of Lyapunov’s theorem.[78]

51.5 Notes[1] Starr (1969)

[2] Howe (1979, p. 1): Howe, Roger (3 November 1979), On the tendency toward convexity of the vector sum of sets (PDF),Cowles Foundation discussion papers 538, Box 2125 Yale Station, New Haven,CT 06520: Cowles Foundation for Researchin Economics, Yale University, retrieved 1 January 2011

[3] Starr (2008)

[4] Guesnerie (1989, p. 138)

[5] (Ekeland 1999, pp. 357–359): Published in the first English edition of 1976, Ekeland’s appendix proves the Shapley–Folkman lemma, also acknowledging Lemaréchal's experiments on page 373.

[6] Bertsekas (1996, pp. 364–381) acknowledging Ekeland (1999) on page 374 and Aubin & Ekeland (1976) on page 381:Bertsekas, Dimitri P. (1996). “5.6 Large scale separable integer programming problems and the exponential method ofmultipliers”. Constrained optimization and Lagrange multiplier methods (Reprint of (1982) Academic Press ed.). Belmont,MA: Athena Scientific. pp. xiii+395. ISBN 1-886529-04-3. MR 690767.Bertsekas (1996, pp. 364–381) describes an application of Lagrangian dual methods to the scheduling of electrical powerplants ("unit commitment problems"), where non-convexity appears because of integer constraints:Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R., Jr.; Posbergh, Thomas A. (January 1983). “Optimal short-termscheduling of large-scale power systems” (PDF). IEEE Transactions on Automatic Control. AC-28 (Proceedings of 1981IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443): 1–11. Retrieved 2 February2011.

[7] Artstein & Vitale (1975, pp. 881–882): Artstein, Zvi; Vitale, Richard A. (1975), “A strong law of large numbers forrandom compact sets”, The Annals of Probability 3 (5): 879–882, doi:10.1214/aop/1176996275, JSTOR 2959130, MR385966, Zbl 0313.60012, PE euclid.ss/1176996275

[8] Carter (2001, p. 94)

[9] Arrow & Hahn (1980, p. 375)

[10] Rockafellar (1997, p. 10)

[11] Arrow & Hahn (1980, p. 376), Rockafellar (1997, pp. 10–11), and Green & Heller (1981, p. 37)

[12] Arrow & Hahn (1980, p. 385) and Rockafellar (1997, pp. 11–12)

[13] Schneider (1993, p. xi) and Rockafellar (1997, p. 16)

[14] Rockafellar (1997, p. 17) and Starr (1997, p. 78)

[15] Schneider (1993, pp. 2–3)

[16] Arrow & Hahn (1980, p. 387)

[17] Starr (1969, pp. 35–36)

[18] Schneider (1993, p. 131)

192 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

[19] Schneider (1993, p. 140) credits this result to Borwein & O'Brien (1978): Borwein, J. M.; O'Brien, R. C. (1978). “Cancel-lation characterizes convexity”. Nanta Mathematica (Nanyang University) 11: 100–102. ISSN 0077-2739. MR 510842.

[20] Schneider (1993, p. 129)

[21] Starr (1969, p. 36)

[22] Starr (1969, p. 37)

[23] Schneider (1993, pp. 129–130)

[24] Arrow & Hahn (1980, pp. 392–395)

[25] Cassels (1975, pp. 435–436)

[26] Schneider (1993, p. 128)

[27] Ekeland (1999, pp. 357–359)

[28] Artstein (1980, p. 180)

[29] Anderson, Robert M. (14 March 2005), “1 The Shapley–Folkman theorem”, Economics 201B: Nonconvex preferences andapproximate equilibria (PDF), Berkeley, CA: Economics Department, University of California, Berkeley, pp. 1–5, retrieved1 January 2011

[30] Starr, Ross M. (1981). “Approximation of points of convex hull of a sum of sets by points of the sum: An elementaryapproach”. Journal of Economic Theory 25 (2): 314–317. doi:10.1016/0022-0531(81)90010-7. MR 640201.

[31] Bertsekas, Dimitri P. (2009). Convex Optimization Theory. Belmont, MA.: Athena Scientific. ISBN 978-1-886529-31-1.

[32] Mas-Colell (1985, pp. 58–61) and Arrow & Hahn (1980, pp. 76–79)

[33] Arrow & Hahn (1980, pp. 79–81)

[34] Starr (1969, p. 26): “After all, one may be indifferent between an automobile and a boat, but in most cases one can neitherdrive nor sail the combination of half boat, half car.”

[35] Hotelling (1935, p. 74): Hotelling, Harold (January 1935). “Demand functions with limited budgets”. Econometrica 3 (1):66–78. JSTOR 1907346.

[36] Wold (1943b, pp. 231 and 239–240): Wold, Herman (1943b). “A synthesis of pure demand analysis II". SkandinaviskAktuarietidskrift [Scandinavian Actuarial Journal] 26: 220–263. MR 11939.Wold & Juréen (1953, p. 146): Wold, Herman; Juréen, Lars (in association with Wold) (1953). “8 Some further appli-cations of preference fields (pp. 129–148)". Demand analysis: A study in econometrics. Wiley publications in statistics.New York: John Wiley and Sons, Inc. pp. xvi+358. MR 64385.

[37] Samuelson (1950, pp. 359–360):

It will be noted that any point where the indifference curves are convex rather than concave cannot beobserved in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumera monopsonist and let him choose between goods lying on a very convex “budget curve” (along which he isaffecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man’sindifference curve from the slope of the observed constraint at the equilibrium point.

Samuelson, Paul A. (November 1950). “The problem of integrability in utility theory”. Economica. New Series 17 (68):355–385. doi:10.2307/2549499. JSTOR 2549499. MR 43436.“Eternal darkness” describes the Hell of John Milton's Paradise Lost, whose concavity is compared to the Serbonian Bogin Book II, lines 592–594:

A gulf profound as that Serbonian BogBetwixt Damiata and Mount Casius old,Where Armies whole have sunk.

Milton’s description of concavity serves as the literary epigraph prefacing chapter seven of Arrow & Hahn (1980, p. 169),“Markets with non-convex preferences and production”, which presents the results of Starr (1969).

[38] Diewert (1982, pp. 552–553)

51.5. NOTES 193

[39] Farrell, M. J. (August 1959). “The Convexity assumption in the theory of competitive markets”. The Journal of PoliticalEconomy 67 (4): 371–391. doi:10.1086/258197. JSTOR 1825163. Farrell, M. J. (October 1961a). “On Convexity,efficiency, and markets: A Reply” 69 (5). pp. 484–489. JSTOR 1828538. Farrell, M. J. (October 1961b). “The Convexityassumption in the theory of competitive markets: Rejoinder” 69 (5). p. 493. JSTOR 1828541.

[40] Bator, Francis M. (October 1961a). “On convexity, efficiency, and markets”. The Journal of Political Economy 69 (5):480–483. doi:10.1086/258540. JSTOR 1828537. Bator, Francis M. (October 1961b). “On convexity, efficiency, andmarkets: Rejoinder” 69 (5). p. 489. JSTOR 1828539.

[41] Koopmans, Tjalling C. (October 1961). “Convexity assumptions, allocative efficiency, and competitive equilibrium”. TheJournal of Political Economy 69 (5): 478–479. doi:10.1086/258539. JSTOR 1828536.Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator(1961a, pp. 482–483), Rothenberg (1960, p. 438), and Starr (1969, p. 26)—commented on Koopmans (1957, pp. 1–126,especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality],and 35–37 [1.7 The role of convexity assumptions in the analysis]):Koopmans, Tjalling C. (1957). “Allocation of resources and the price system”. In Koopmans, Tjalling C. Three essays onthe state of economic science. New York: McGraw–Hill Book Company. pp. 1–126. ISBN 0-07-035337-9.

[42] Rothenberg (1960, p. 447): Rothenberg, Jerome (October 1960). “Non-convexity, aggregation, and Pareto optimality”.The Journal of Political Economy 68 (5): 435–468. doi:10.1086/258363. JSTOR 1830308. (Rothenberg, Jerome (October1961). “Comments on non-convexity” 69 (5). pp. 490–492. JSTOR 1828540.)

[43] Arrow & Hahn (1980, p. 182)

[44] Shapley & Shubik (1966, p. 806): Shapley, L. S.; Shubik, M. (October 1966). “Quasi-cores in a monetary economy withnonconvex preferences”. Econometrica 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 154.45303.

[45] Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). “Existence of competitive equilibrium in markets with acontinuum of traders”. Econometrica 34 (1): 1–17. JSTOR 1909854. MR 191623. Aumann (1966) uses results fromAumann (1964, 1965):Aumann, Robert J. (January–April 1964). “Markets with a continuum of traders”. Econometrica 32 (1–2): 39–50. JSTOR1913732. MR 172689.Aumann, Robert J. (August 1965). “Integrals of set-valued functions”. Journal of Mathematical Analysis and Applications12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 185073.

[46] Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold &Juréen (1953, p. 146), according to Diewert (1982, p. 552).

[47] Starr & Stinchcombe (1999, pp. 217–218): Starr, R. M.; Stinchcombe, M. B. (1999). “Exchange in a network of tradingposts”. In Chichilnisky, Graciela. Markets, information and uncertainty: Essays in economic theory in honor of Kenneth J.Arrow. Cambridge: Cambridge University Press. pp. 217–234. doi:10.2277/0521553555. ISBN 978-0-521-08288-4.

[48] Arrow & Hahn (1980, pp. 169–182). Starr (1969, pp. 27–33)

[49] Green & Heller (1981, p. 44)

[50] Guesnerie (1989, pp. 99)

[51] Mas-Colell (1987)

[52] Varian (1992, pp. 393–394): Varian, Hal R. (1992). “21.2 Convexity and size”. Microeconomic Analysis (3rd ed.). W.W. Norton & Company. ISBN 978-0-393-95735-8. MR 1036734.Mas-Colell, Whinston & Green (1995, pp. 627–630): Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995).“17.1 Large economies and nonconvexities”. Microeconomic theory. Oxford University Press. ISBN 978-0-19-507340-9.

[53] Arrow & Hahn (1980, pp. 169–182)Mas-Colell (1985, pp. 52–55, 145–146, 152–153, and 274–275): Mas-Colell, Andreu (1985). “1.L Averages of sets”.The Theory of general economic equilibrium: A differentiable approach. Econometric Society monographs 9. CambridgeUniversity Press. ISBN 0-521-26514-2. MR 1113262.Hildenbrand (1974, pp. 37, 115–116, 122, and 168): Hildenbrand, Werner (1974). Core and equilibria of a large economy.Princeton studies in mathematical economics 5. Princeton, N.J.: Princeton University Press. pp. viii+251. ISBN 978-0-691-04189-6. MR 389160.

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[54] Starr (1997, p. 169): Starr, Ross M. (1997). “8 Convex sets, separation theorems, and non-convex sets inRN (new chapters22 and 25–26 in (2011) second ed.)". General equilibrium theory: An introduction (First ed.). Cambridge: CambridgeUniversity Press. pp. xxiii+250. ISBN 0-521-56473-5. MR 1462618.Ellickson (1994, pp. xviii, 306–310, 312, 328–329, 347, and 352): Ellickson, Bryan (1994). Competitive equilibrium:Theory and applications. Cambridge University Press. doi:10.2277/0521319889. ISBN 978-0-521-31988-1.

[55] Laffont (1988, pp. 63–65): Laffont, Jean-Jacques (1988). “3 Nonconvexities”. 0-262-12127-1&id=O5MnAQAAIAAJFundamentals of public economics. MIT. ISBN 0-262-12127-1.

[56] Salanié (2000, pp. 112–113 and 107–115): Salanié, Bernard (2000). “7 Nonconvexities”. Microeconomics of marketfailures (English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris) ed.).Cambridge, MA: MIT Press. pp. 107–125. ISBN 0-262-19443-0.

[57] Ichiishi (1983, pp. 24–25): Ichiishi, Tatsuro (1983). Game theory for economic analysis. Economic theory, econometrics,and mathematical economics. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. x+164.ISBN 0-12-370180-5. MR 700688.

[58] Cassels (1981, pp. 127 and 33–34): Cassels, J. W. S. (1981). “Appendix A Convex sets”. Economics for mathematicians.London Mathematical Society lecture note series 62. Cambridge, New York: Cambridge University Press. pp. xi+145.ISBN 0-521-28614-X. MR 657578.

[59] Aubin (2007, pp. 458–476): Aubin, Jean-Pierre (2007). “14.2 Duality in the case of non-convex integral criterion andconstraints (especially 14.2.3 The Shapley–Folkman theorem, pages 463–465)". Mathematical methods of game and eco-nomic theory (Reprint with new preface of 1982 North-Holland revised English ed.). Mineola, NY: Dover Publications,Inc. pp. xxxii+616. ISBN 978-0-486-46265-3. MR 2449499.

[60] Carter (2001, pp. 93–94, 143, 318–319, 375–377, and 416)

[61] Trockel (1984, p. 30): Trockel, Walter (1984). Market demand: An analysis of large economies with nonconvex preferences.Lecture notes in economics and mathematical systems 223. Berlin: Springer-Verlag. pp. viii+205. ISBN 3-540-12881-6.MR 737006.

[62] Bertsekas (1999, p. 496): Bertsekas, Dimitri P. (1999). “5.1.6 Separable problems and their geometry”. NonlinearProgramming (Second ed.). Cambridge, MA.: Athena Scientific. pp. 494–498. ISBN 1-886529-00-0.

[63] Rockafellar (1997, p. 23)

[64] The limit of a sequence is a member of the closure of the original set, which is the smallest closed set that contains theoriginal set. The Minkowski sum of two closed sets need not be closed, so the following inclusion can be strict

Clos(P) + Clos(Q) ⊆ Clos( Clos(P) + Clos(Q) );

the inclusion can be strict even for two convex closed summand-sets, according to Rockafellar (1997, pp. 49 and 75).Ensuring that the Minkowski sum of sets be closed requires the closure operation, which appends limits of convergentsequences.

[65] Lemaréchal (1973, p. 38): Lemaréchal, Claude (April 1973), Utilisation de la dualité dans les problémes non convexes [Useof duality for non–convex problems] (in French) (16), Domaine de Voluceau, Rocquencourt, 78150 Le Chesnay, France:IRIA (now INRIA), Laboratoire de recherche en informatique et automatique, p. 41. Lemaréchal’s experiments werediscussed in later publications:Aardal (1995, pp. 2–3): Aardal, Karen (March 1995). "Optima interview Claude Lemaréchal” (PDF). Optima: Mathe-matical Programming Society newsletter 45: 2–4. Retrieved 2 February 2011.Hiriart-Urruty & Lemaréchal (1993, pp. 143–145, 151, 153, and 156): Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude(1993). “XII Abstract duality for practitioners”. Convex analysis and minimization algorithms, Volume II: Advanced the-ory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences] 306. Berlin: Springer-Verlag. pp. 136–193 (and bibliographical comments on pp. 334–335). ISBN 3-540-56852-2. MR 1295240.

[66] Ekeland, Ivar (1974). “Une estimation a priori en programmation non convexe”. Comptes Rendus Hebdomadaires desSéances de l'Académie des Sciences. Séries A et B (in French) 279: 149–151. ISSN 0151-0509. MR 395844.

[67] Aubin & Ekeland (1976, pp. 226, 233, 235, 238, and 241): Aubin, J. P.; Ekeland, I. (1976). “Estimates of the dualitygap in nonconvex optimization”. Mathematics of Operations Research 1 (3): 225–245. doi:10.1287/moor.1.3.225. JSTOR3689565. MR 449695.Aubin & Ekeland (1976) and Ekeland (1999, pp. 362–364) also considered the convex closure of a problem of non-convexminimization—that is, the problem defined as the closed convex hull of the epigraph of the original problem. Their study

51.6. REFERENCES 195

of duality gaps was extended by Di Guglielmo to the quasiconvex closure of a non-convex minimization problem—that is,the problem defined as the closed convex hull of the lower level sets:Di Guglielmo (1977, pp. 287–288): Di Guglielmo, F. (1977). “Nonconvex duality in multiobjective optimization”. Math-ematics of Operations Research 2 (3): 285–291. doi:10.1287/moor.2.3.285. JSTOR 3689518. MR 484418.

[68] Schneider & Weil (2008, p. 45): Schneider, Rolf; Weil, Wolfgang (2008). Stochastic and integral geometry. Probabilityand its applications. Springer. doi:10.1007/978-3-540-78859-1. ISBN 978-3-540-78858-4. MR 2455326.

[69] Cassels (1975, pp. 433–434): Cassels, J. W. S. (1975). “Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem”. Mathematical Proceedings of the Cambridge Philosophical Society 78 (3): 433–436. doi:10.1017/S0305004100051884.MR 385711.

[70] Molchanov (2005, pp. 195–198, 218, 232, 237–238 and 407): Molchanov, Ilya (2005). “3 Minkowski addition”. Theoryof random sets. Probability and its applications. London: Springer-Verlag London Ltd. pp. 194–240. doi:10.1007/1-84628-150-4. ISBN 978-1-84996-949-9. MR 2132405.

[71] Puri & Ralescu (1985, pp. 154–155): Puri, Madan L.; Ralescu, Dan A. (1985). “Limit theorems for random compact setsin Banach space”. Mathematical Proceedings of the Cambridge Philosophical Society 97 (1): 151–158. doi:10.1017/S0305004100062691.MR 764504.

[72] Weil (1982, pp. 203, and 205–206): Weil, Wolfgang (1982). “An application of the central limit theorem for Banach-space–valued random variables to the theory of random sets”. Zeitschrift für Wahrscheinlichkeitstheorie und VerwandteGebiete [Probability Theory and Related Fields] 60 (2): 203–208. doi:10.1007/BF00531823. MR 663901.

[73] Cerf (1999, pp. 243–244): Cerf, Raphaël (1999). “Large deviations for sums of i.i.d. random compact sets”. Proceedingsof the American Mathematical Society 127 (8): 2431–2436. doi:10.1090/S0002-9939-99-04788-7. MR 1487361. Cerfuses applications of the Shapley–Folkman lemma from Puri & Ralescu (1985, pp. 154–155).

[74] Ruzsa (1997, p. 345): Ruzsa, Imre Z. (1997). “The Brunn–Minkowski inequality and nonconvex sets”. GeometriaeDedicata 67 (3): 337–348. doi:10.1023/A:1004958110076. MR 1475877.

[75] Tardella (1990, pp. 478–479): Tardella, Fabio (1990). “A new proof of the Lyapunov convexity theorem”. SIAM Journalon Control and Optimization 28 (2): 478–481. doi:10.1137/0328026. MR 1040471.

[76] Vind (1964, pp. 168 and 175): Vind, Karl (May 1964). “Edgeworth-allocations in an exchange economy with manytraders”. International Economic Review 5 (2): 165–77. JSTOR 2525560. Vind’s article was noted by the winner of the1983 Nobel Prize in Economics, Gérard Debreu. Debreu (1991, p. 4) wrote:

The concept of a convex set (i.e., a set containing the segment connecting any two of its points) hadrepeatedly been placed at the center of economic theory before 1964. It appeared in a new light with theintroduction of integration theory in the study of economic competition: If one associates with every agent ofan economy an arbitrary set in the commodity space and if one averages those individual sets over a collectionof insignificant agents, then the resulting set is necessarily convex. [Debreu appends this footnote: “On thisdirect consequence of a theorem of A. A. Lyapunov, see Vind (1964)."] But explanations of the ... functionsof prices ... can be made to rest on the convexity of sets derived by that averaging process. Convexity in thecommodity space obtained by aggregation over a collection of insignificant agents is an insight that economictheory owes ... to integration theory. [Italics added]

Debreu, Gérard (March 1991). “The Mathematization of economic theory”. The American Economic Review 81 (Presi-dential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington,DC): 1–7. JSTOR 2006785.

[77] Artstein (1980, pp. 172–183) Artstein (1980) was republished in a festschrift for Robert J. Aumann, winner of the 2008Nobel Prize in Economics: Artstein, Zvi (1995). “22 Discrete and continuous bang–bang and facial spaces or: Look forthe extreme points”. In Hart, Sergiu; Neyman, Abraham. Game and economic theory: Selected contributions in honor ofRobert J. Aumann. Ann Arbor, MI: University of Michigan Press. pp. 449–462. ISBN 0-472-10673-2.

[78] Mas-Colell (1978, p. 210): Mas-Colell, Andreu (1978). “A note on the core equivalence theorem: How many blockingcoalitions are there?". Journal of Mathematical Economics 5 (3): 207–215. doi:10.1016/0304-4068(78)90010-1. MR514468.

51.6 References• Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in

Economics 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Ams-terdam: North-Holland. ISBN 0-444-85497-5. MR 439057.

196 CHAPTER 51. SHAPLEY–FOLKMAN LEMMA

• Artstein, Zvi (1980). “Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points”.SIAM Review 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 564562.

• Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. pp. xx+649.ISBN 0-262-53192-5. MR 1865841. (Author’s website with answers to exercises).

• Diewert, W. E. (1982). “12 Duality approaches to microeconomic theory”. In Arrow, Kenneth Joseph; Intrili-gator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics 1. Amsterdam:North-Holland Publishing Co. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 648778.

• Ekeland, Ivar (1999) [1976]. “Appendix I: An a priori estimate in convex programming”. In Ekeland, Ivar;Temam, Roger. Convex analysis and variational problems. Classics in Applied Mathematics 28 (Correctedreprinting of the North-Holland ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics(SIAM). pp. 357–373. ISBN 0-89871-450-8. MR 1727362.

• Green, Jerry; Heller, Walter P. (1981). “1 Mathematical analysis and convexity with applications to eco-nomics”. In Arrow, Kenneth Joseph; Intriligator, Michael D. Handbook of mathematical economics, VolumeI. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 634800.

• Guesnerie, Roger (1989). “First-best allocation of resources with nonconvexities in production”. In Cornet,Bernard; Tulkens, Henry. Contributions to Operations Research and Economics: The twentieth anniversary ofCORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, MA: MIT Press.pp. 99–143. ISBN 0-262-03149-3. MR 1104662.

• Mas-Colell, A. (1987). “Non-convexity”. In Eatwell, John; Milgate, Murray; Newman, Peter. The new Pal-grave: A dictionary of economics (first ed.). Palgrave Macmillan. pp. 653–661. doi:10.1057/9780230226203.3173.(PDF file at Mas-Colell’s homepage).

• Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970(MR 274683) Princeton Mathematical Series 28 ed.). Princeton, NJ: Princeton University Press. pp. xviii+451.ISBN 0-691-01586-4. MR 1451876.

• Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and itsApplications 44. Cambridge: Cambridge University Press. pp. xiv+490. ISBN 0-521-35220-7. MR 1216521.

• Starr, Ross M. (1969), “Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37)", Econometrica 37 (1): 25–38, JSTOR 1909201

• Starr, Ross M. (2008). “Shapley–Folkman theorem”. In Durlauf, Steven N.; Blume, Lawrence E. The new Pal-grave dictionary of economics (Second ed.). Palgrave Macmillan. pp. 317–318 (1st ed.). doi:10.1057/9780230226203.1518.

51.7 External links• Anderson, Robert M. (March 2005), “1 The Shapley–Folkman theorem”, Economics 201B: Nonconvex pref-erences and approximate equilibria (PDF), Berkeley, CA: Economics Department, University of California,Berkeley, pp. 1–5, retrieved 15 January 2011

• Howe, Roger (November 1979), On the tendency toward convexity of the vector sum of sets (PDF), CowlesFoundation discussion papers 538, Box 2125 Yales Station, New Haven, CT 06520: Cowles Foundation forResearch in Economics, Yale University, retrieved 15 January 2011

• Starr, Ross M. (September 2009), “8 Convex sets, separation theorems, and non-convex sets in RN (Section8.2.3 Measuring non-convexity, the Shapley–Folkman theorem)", General equilibrium theory: An introduction(PDF), pp. 3–6, MR 1462618, (Draft of second edition, from Starr’s course at the Economics Department ofthe University of California, San Diego), retrieved 15 January 2011

• Starr, Ross M. (May 2007), Shapley–Folkman theorem (PDF), pp. 1–3, (Draft of article for the second editionof New Palgrave Dictionary of Economics), retrieved 15 January 2011

Chapter 52

Sigma-algebra

"Σ-algebra” redirects here. For an algebraic structure admitting a given signature Σ of operations, see Universal al-gebra.

In mathematical analysis and in probability theory, a σ-algebra (also sigma-algebra, σ-field, sigma-field) on a setX is a collection Σ of subsets of X that is closed under countable-fold set operations (complement, union of countablymany sets and intersection of countably many sets). By contrast, an algebra is only required to be closed under finitelymany set operations. That is, a σ-algebra is an algebra of sets, completed to include countably infinite operations.The pair (X, Σ) is also a field of sets, called a measurable space.The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for whicha given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as thefoundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events whichcan be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.In statistics, (sub) σ-algebras are needed for a formal mathematical definition of sufficient statistic,[1] particularlywhen the statistic is a function or a random process and the notion of conditional density is not applicable.If X = a, b, c, d, one possible σ-algebra on X is Σ = ∅, a, b, c, d, a, b, c, d , where ∅ is the empty set.However, a finite algebra is always a σ-algebra.If A1, A2, A3, … is a countable partition of X then the collection of all unions of sets in the partition (includingthe empty set) is a σ-algebra.A more useful example is the set of subsets of the real line formed by starting with all open intervals and addingin all countable unions, countable intersections, and relative complements and continuing this process (by transfiniteiteration through all countable ordinals) until the relevant closure properties are achieved (a construction known asthe Borel hierarchy).

52.1 Motivation

There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managingpartial information characterized by sets.

52.1.1 Measure

A measure on X is a function that assigns a non-negative real number to subsets of X; this can be thought of as makingprecise a notion of “size” or “volume” for sets. We want the size of the union of disjoint sets to be the sum of theirindividual sizes, even for an infinite sequence of disjoint sets.One would like to assign a size to every subset of X, but in many natural settings, this is not possible. For example theaxiom of choice implies that when the size under consideration is the ordinary notion of length for subsets of the realline, then there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead asmaller collection of privileged subsets of X. These subsets will be called the measurable sets. They are closed under

197

198 CHAPTER 52. SIGMA-ALGEBRA

operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable setand the countable union of measurable sets is a measurable set. Non-empty collections of sets with these propertiesare called σ-algebras.

52.1.2 Limits of sets

Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences ofsets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows onσ-algebras.

• The limit supremum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim supn→∞

An =∞∩

n=1

∞∪m=n

Am.

• The limit infimum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim infn→∞

An =∞∪

n=1

∞∩m=n

Am.

• If, in fact,

lim infn→∞

An = lim supn→∞

An

then the limn→∞ An exists as that common set.

52.1.3 Sub σ-algebras

In much of probability, especially when conditional expectation is involved, one is concerned with sets that representonly part of all the possible information that can be observed. This partial information can be characterized with asmaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only toand determined only by the partial information. A simple example suffices to illustrate this idea.Imagine you are playing a game that involves flipping a coin repeatedly and observing whether it comes up Heads (H)or Tails (T). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last.This means the sample space Ω must consist of all possible infinite sequences of H or T :

Ω = H,T∞ = (x1, x2, x3, . . . ) : xi ∈ H,T, i ≥ 1

However, after n flips of the coin, you may want to determine or revise your betting strategy in advance of the nextflip. The observed information at that point can be described in terms of the 2n possibilities for the first n flips.Formally, since you need to use subsets of Ω, this is codified as the σ-algebra

Gn = A× H,T∞ : A ⊂ H,Tn

Observe that then

G1 ⊂ G2 ⊂ G3 ⊂ · · · ⊂ G∞

where G∞ is the smallest σ-algebra containing all the others.

52.2. DEFINITION AND PROPERTIES 199

52.2 Definition and properties

52.2.1 Definition

Let X be some set, and let 2X represent its power set. Then a subset Σ ⊂ 2X is called a σ-algebra if it satisfies thefollowing three properties:[2]

1. X is in Σ.

2. Σ is closed under complementation: If A is in Σ, then so is its complement, X\A.

3. Σ is closed under countable unions: If A1, A2, A3, ... are in Σ, then so is A = A1 ∪ A2 ∪ A3 ∪ … .

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying DeMorgan’s laws).It also follows that the empty set ∅ is in Σ, since by (1) X is in Σ and (2) asserts that its complement, the empty set,is also in Σ. Moreover, by (3) it follows as well that X, ∅ is the smallest possible σ-algebra.Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-algebraover X, is called a measurable space. A function between two measurable spaces is called a measurable function ifthe preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with themeasurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to [0, ∞].A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin’s theorem(below).

52.2.2 Dynkin’s π-λ theorem

This theorem (or the related monotone class theorem) is an essential tool for proving many results about propertiesof specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.

A π-system P is a collection of subsets of Σ that is closed under finitely many intersections, anda Dynkin system (or λ-system) D is a collection of subsets of Σ that contains Σ and is closed undercomplement and under countable unions of disjoint subsets.

Dynkin’s π-λ theorem says, if P is a π-system and D is a Dynkin system that contains P then the σ-algebra σ(P)generated by P is contained in D. Since certain π-systems are relatively simple classes, it may not be hard to verifythat all sets in P enjoy the property under consideration while, on the other hand, showing that the collection D of allsubsets with the property is a Dynkin system can also be straightforward. Dynkin’s π-λ Theorem then implies thatall sets in σ(P) enjoy the property, avoiding the task of checking it for an arbitrary set in σ(P).One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures orintegrals. For example, it is used to equate a probability for a random variable X with the Lebesgue-Stieltjes integraltypically associated with computing the probability:

P(X ∈ A) =∫AF (dx) for all A in the Borel σ-algebra on R,

where F(x) is the cumulative distribution function for X, defined on R, while P is a probability measure, defined ona σ-algebra Σ of subsets of some sample space Ω.

52.2.3 Combining σ-algebras

Suppose Σα : α ∈ A is a collection of σ-algebras on a space X.

• The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it oftenis denoted by:

200 CHAPTER 52. SIGMA-ALGEBRA

∧α∈A

Σα.

Sketch of Proof: Let Σ∗ denote the intersection. Since X is in every Σα, Σ∗ is not empty. Closure undercomplement and countable unions for every Σα implies the same must be true for Σ∗. Therefore, Σ∗ is aσ-algebra.

• The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates aσ-algebra known as the join which typically is denoted

∨α∈A

Σα = σ

( ∪α∈A

Σα

).

A π-system that generates the join is

P =

n∩

i=1

Ai : Ai ∈ Σαi , αi ∈ A, n ≥ 1

.

Sketch of Proof: By the case n = 1, it is seen that each Σα ⊂ P , so∪α∈A

Σα ⊂ P.

This implies

σ

( ∪α∈A

Σα

)⊂ σ(P)

by the definition of a σ-algebra generated by a collection of subsets. On the other hand,

P ⊂ σ

( ∪α∈A

Σα

)

which, by Dynkin’s π-λ theorem, implies

σ(P) ⊂ σ

( ∪α∈A

Σα

).

52.2.4 σ-algebras for subspaces

Suppose Y is a subset of X and let (X, Σ) be a measurable space.

• The collection Y ∩ B: B ∈ Σ is a σ-algebra of subsets of Y.

• Suppose (Y, Λ) is a measurable space. The collection A ⊂ X : A ∩ Y ∈ Λ is a σ-algebra of subsets of X.

52.2.5 Relation to σ-ring

A σ-algebra Σ is just a σ-ring that contains the universal set X.[3] A σ-ring need not be a σ-algebra, as for examplemeasurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real linehas infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takesmeasurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtainedby their countable union yet its measure is not finite.

52.3. EXAMPLES 201

52.2.6 Typographic note

σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus (X, Σ) may bedenoted as (X,F) or (X,F) .

52.3 Examples

52.3.1 Simple set-based examples

Let X be any set.

• The family consisting only of the empty set and the set X, called the minimal or trivial σ-algebra over X.

• The power set of X, called the discrete σ-algebra.

• The collection ∅, A, Ac, X is a simple σ-algebra generated by the subset A.

• The collection of subsets of X which are countable or whose complements are countable is a σ-algebra (whichis distinct from the power set of X if and only if X is uncountable). This is the σ-algebra generated by thesingletons of X. Note: “countable” includes finite or empty.

• The collection of all unions of sets in a countable partition of X is a σ-algebra.

52.3.2 Stopping time sigma-algebras

A stopping time τ can define a σ -algebraFτ , the so-called stopping time sigma-algebra, which in a filtered probabilityspace describes the information up to the random time τ in the sense that, if the filtered probability space is interpretedas a random experiment, the maximum information that can be found out about the experiment from arbitrarily oftenrepeating it until the time τ is Fτ .[4]

52.4 σ-algebras generated by families of sets

52.4.1 σ-algebra generated by an arbitrary family

Let F be an arbitrary family of subsets of X. Then there exists a unique smallest σ-algebra which contains every setin F (even though F may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containingF. (See intersections of σ-algebras above.) This σ-algebra is denoted σ(F) and is called the σ-algebra generated byF.For a simple example, consider the set X = 1, 2, 3. Then the σ-algebra generated by the single subset 1 isσ(1) = ∅, 1, 2, 3, 1, 2, 3. By an abuse of notation, when a collection of subsets contains only oneelement, A, one may write σ(A) instead of σ(A); in the prior example σ(1) instead of σ(1). Indeed, usingσ(A1, A2, ...) to mean σ(A1, A2, ...) is also quite common.There are many families of subsets that generate useful σ-algebras. Some of these are presented here.

52.4.2 σ-algebra generated by a function

If f is a function from a set X to a set Y and B is a σ-algebra of subsets of Y, then the σ-algebra generated by thefunction f, denoted by σ(f), is the collection of all inverse images f−1(S) of the sets S in B. i.e.

σ(f) = f−1(S) |S ∈ B.

A function f from a set X to a set Y is measurable with respect to a σ-algebra Σ of subsets of X if and only if σ(f) isa subset of Σ.

202 CHAPTER 52. SIGMA-ALGEBRA

One common situation, and understood by default if B is not specified explicitly, is when Y is a metric or topologicalspace and B is the collection of Borel sets on Y.If f is a function from X to Rn then σ(f) is generated by the family of subsets which are inverse images of inter-vals/rectangles in Rn:

σ(f) = σ(f−1((a1, b1]× · · · × (an, bn]) : ai, bi ∈ R

).

A useful property is the following. Assume f is a measurable map from (X, ΣX) to (S, ΣS) and g is a measurable mapfrom (X, ΣX) to (T, ΣT). If there exists a measurable function h from T to S such that f(x) = h(g(x)) then σ(f) ⊂σ(g). If S is finite or countably infinite or if (S, ΣS) is a standard Borel space (e.g., a separable complete metric spacewith its associated Borel sets) then the converse is also true.[5] Examples of standard Borel spaces include Rn with itsBorel sets and R∞ with the cylinder σ-algebra described below.

52.4.3 Borel and Lebesgue σ-algebras

An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or,equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivialexample that is not a Borel set, see the Vitali set or Non-Borel sets.On the Euclidean space Rn, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebracontains more sets than the Borel σ-algebra onRn and is preferred in integration theory, as it gives a complete measurespace.

52.4.4 Product σ-algebra

Let (X1,Σ1) and (X2,Σ2) be two measurable spaces. The σ-algebra for the corresponding product space X1 ×X2

is called the product σ-algebra and is defined by

Σ1 × Σ2 = σ(B1 ×B2 : B1 ∈ Σ1, B2 ∈ Σ2).

Observe that B1 ×B2 : B1 ∈ Σ1, B2 ∈ Σ2 is a π-system.The Borel σ-algebra for Rn is generated by half-infinite rectangles and by finite rectangles. For example,

B(Rn) = σ ((−∞, b1]× · · · × (−∞, bn] : bi ∈ R) = σ ((a1, b1]× · · · × (an, bn] : ai, bi ∈ R) .

For each of these two examples, the generating family is a π-system.

52.4.5 σ-algebra generated by cylinder sets

Suppose

X ⊂ RT = f : f(t) ∈ R, t ∈ T

is a set of real-valued functions. Let B(R) denote the Borel subsets of R. A cylinder subset of X is a finitely restrictedset defined as

Ct1,...,tn(B1, . . . , Bn) = f ∈ X : f(ti) ∈ Bi, 1 ≤ i ≤ n.

Each

Ct1,...,tn(B1, . . . , Bn) : Bi ∈ B(R), 1 ≤ i ≤ n

52.5. SEE ALSO 203

is a π-system that generates a σ-algebra Σt1,...,tn . Then the family of subsets

FX =

∞∪n=1

∪ti∈T,i≤n

Σt1,...,tn

is an algebra that generates the cylinder σ-algebra for X. This σ-algebra is a subalgebra of the Borel σ-algebradetermined by the product topology of RT restricted to X.An important special case is when T is the set of natural numbers and X is a set of real-valued sequences. In thiscase, it suffices to consider the cylinder sets

Cn(B1, . . . , Bn) = (B1 × · · · ×Bn × R∞) ∩X = (x1, x2, . . . , xn, xn+1, . . . ) ∈ X : xi ∈ Bi, 1 ≤ i ≤ n,

for which

Σn = σ(Cn(B1, . . . , Bn) : Bi ∈ B(R), 1 ≤ i ≤ n)

is a non-decreasing sequence of σ-algebras.

52.4.6 σ-algebra generated by random variable or vector

Suppose (Ω,Σ,P) is a probability space. If Y : Ω → Rn is measurable with respect to the Borel σ-algebra on Rn

then Y is called a random variable (n = 1) or random vector (n ≥ 1). The σ-algebra generated by Y is

σ(Y ) = Y −1(A) : A ∈ B(Rn).

52.4.7 σ-algebra generated by a stochastic process

Suppose (Ω,Σ,P) is a probability space and RT is the set of real-valued functions on T . If Y : Ω → X ⊂ RT ismeasurable with respect to the cylinder σ-algebra σ(FX) (see above) for X then Y is called a stochastic process orrandom process. The σ-algebra generated by Y is

σ(Y ) =Y −1(A) : A ∈ σ(FX)

= σ(Y −1(A) : A ∈ FX),

the σ-algebra generated by the inverse images of cylinder sets.

52.5 See also

• Join (sigma algebra)

• Measurable function

• Sample space

• Separable sigma algebra

• Sigma ring

• Sigma additivity

204 CHAPTER 52. SIGMA-ALGEBRA

52.6 References[1] Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1118122372.

[2] Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.

[3] Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 9780470317952.

[4] Fischer, Tom (2013). “On simple representations of stopping times and stopping time sigma-algebras”. Statistics andProbability Letters 83 (1): 345–349. doi:10.1016/j.spl.2012.09.024.

[5] Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 0-387-95313-2.

52.7 External links• Hazewinkel, Michiel, ed. (2001), “Algebra of sets”, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

• Sigma Algebra from PlanetMath.

Chapter 53

Sigma-ideal

In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read “sigma,” means countable in thiscontext) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent applicationis perhaps in probability theory.Let (X,Σ) be a measurable space (meaning Σ is a σ-algebra of subsets of X). A subset N of Σ is a σ-ideal if thefollowing properties are satisfied:(i) Ø ∈ N;(ii) When A ∈ N and B ∈ Σ , B ⊆ A ⇒ B ∈ N;(iii) Ann∈N ⊆ N ⇒

∪n∈N An ∈ N.

Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. Theconcept of σ-ideal is dual to that of a countably complete (σ-) filter.If a measure μ is given on (X,Σ), the set of μ-negligible sets (S ∈ Σ such that μ(S) = 0) is a σ-ideal.The notion can be generalized to preorders (P,≤,0) with a bottom element 0 as follows: I is a σ-ideal of P just when(i') 0 ∈ I,(ii') x ≤ y & y ∈ I ⇒ x ∈ I, and(iii') given a family xn ∈ I (n ∈ N), there is y ∈ I such that xn ≤ y for each nThus I contains the bottom element, is downward closed, and is closed under countable suprema (which must exist).It is natural in this context to ask that P itself have countable suprema.

53.1 References• Bauer, Heinz (2001): Measure and Integration Theory. Walter de Gruyter GmbH & Co. KG, 10785 Berlin,

Germany.

205

Chapter 54

Sigma-ring

In mathematics, a nonempty collection of sets is called a σ-ring (pronounced sigma-ring) if it is closed under countableunion and relative complementation.

54.1 Formal definition

LetR be a nonempty collection of sets. ThenR is a σ-ring if:

1.∪∞

n=1 An ∈ R if An ∈ R for all n ∈ N

2. A \B ∈ R if A,B ∈ R

54.2 Properties

From these two properties we immediately see that

∩∞n=1 An ∈ R if An ∈ R for all n ∈ N

This is simply because ∩∞n=1An = A1 \ ∪∞n=1(A1 \An) .

54.3 Similar concepts

If the first property is weakened to closure under finite union (i.e., A∪B ∈ Rwhenever A,B ∈ R ) but not countableunion, thenR is a ring but not a σ-ring.

54.4 Uses

σ-rings can be used instead of σ-fields (σ-algebras) in the development of measure and integration theory, if one doesnot wish to require that the universal set be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be aσ-field.A σ-ring R that is a collection of subsets of X induces a σ-field for X . Define A to be the collection of all subsetsof X that are elements ofR or whose complements are elements ofR . Then A is a σ-field over the set X . In factA is the minimal σ-field containingR since it must be contained in every σ-field containingR .

206

54.5. SEE ALSO 207

54.5 See also• Delta ring

• Ring of sets

• Sigma field

54.6 References• Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses σ-rings

in development of Lebesgue theory.

Chapter 55

Simplex category

In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the categoryof non-empty finite ordinals and order preserving maps. It is used to define simplicial and cosimplicial objects.

55.1 Formal definition

The simplex category is usually denoted by ∆ . There are several equivalent descriptions of this category. ∆ canbe described as the category of non-empty finite ordinals as objects, thought of as totally ordered sets, and orderpreserving functions as morphisms. The objects are commonly denoted [n] = 0, 1, . . . , n (so that [n] is the ordinaln+1 ). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elementsof the orderings. (See simplicial set for relations of these maps.)A simplicial object is a presheaf on ∆ , that is a contravariant functor from ∆ to another category. For instance,simplicial sets are contravariant with the codomain category being the category of sets. A cosimplicial object isdefined similarly as a covariant functor originating from ∆ .

55.2 Augmented simplex category

The augmented simplex category, denoted by ∆+ is the category of all finite ordinals and order preserving maps,thus ∆+ = ∆∪ [−1] , where [−1] = ∅ . Accordingly, this category might also be denoted FinOrd. The augmentedsimplex category is occasionally referred to as algebraists’ simplex category and the above version is called topologists’simplex category.A contravariant functor defined on ∆+ is called an augmented simplicial object and a covariant functor out of ∆+

is called an augmented cosimplicial object; when the codmain category is the category of sets, for example, theseare called augmented simplicial sets and augmented cosimplicial sets respectively.The augmented simplex category, unlike the simplex category, admits a natural monoidal structure. The monoidalproduct is given by concatenation of linear orders, and the unit is the empty ordinal [−1] (the lack of a unit preventsthis from qualifying as a monoidal structure on ∆ ). In fact, ∆+ is the monoidal category freely generated by asingle monoid object, given by [0] with the unique possible unit and multiplication. This description is useful forunderstanding how any comonoid object in a monoidal category gives rise to a simplicial object since it can thenbe viewed as the image of a functor from ∆op

+ to the monoidal category containing the comonoid; by forgetting theaugmentation we obtain a simplicial object. Similarly, this also illuminates the construction of simplicial sets frommonads (and hence adjoint functors) since monads can be viewed as monoid objects in endofunctor categories.The augmented simplex category provides a simple example of a compact closed category.

55.3 See also

• Simplicial category

208

55.4. REFERENCES 209

• PRO (category theory)

55.4 References• P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics Vol. 174, Birkhäuser

Basel-Boston-Berlin (1999) ISBN 3-7643-6064-X

55.5 External links• Simplex category in nLab

Chapter 56

Simplicial approximation theorem

In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeingthat continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplestkind. It applies to mappings between spaces that are built up from simplices — that is, finite simplicial complexes.The general continuous mapping between such spaces can be represented approximately by the type of mapping thatis (affine-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision of thesimplices of the domain, and (ii) replacement of the actual mapping by a homotopic one.This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based oncompactness). It served to put the homology theory of the time — the first decade of the twentieth century —on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mappings couldin a given case be expressed in a finitary way. This must be seen against the background of a realisation at the timethat continuity was in general compatible with the pathological, in some other areas. This initiated, one could say,the era of combinatorial topology.There is a further simplicial approximation theorem for homotopies, stating that a homotopy between continuousmappings can likewise be approximated by a combinatorial version.

56.1 Formal statement of the theorem

Let K and L be two simplicial complexes. A simplicial mapping f : K → L is called a simplicial approximation ofa continuous function F : |K| → |L| if for every point x ∈ |K| , |f |(x) belongs to the minimal closed simplex of Lcontaining the point F (x) . If f is a simplicial approximation to a continuous map F , then the geometric realizationof f , |f | is necessarily homotopic to F .The simplicial approximation theorem states that given any continuous map F : |K| → |L| there exists a naturalnumber n0 such that for all n ≥ n0 there exists a simplicial approximation f : BdnK → L to F (where Bd Kdenotes the barycentric subdivision of K , and BdnK denotes the result of applying barycentric subdivision n times.)

56.2 References• Hazewinkel, Michiel, ed. (2001), “Simplicial complex”, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

210

Chapter 57

Simplicial complex

A simplicial 3-complex.

In mathematics, a simplicial complex is a topological space of a certain kind, constructed by “gluing together” points,line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not beconfused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. Thepurely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.

211

212 CHAPTER 57. SIMPLICIAL COMPLEX

57.1 Definitions

A simplicial complex K is a set of simplices that satisfies the following conditions:

1. Any face of a simplex from K is also in K .2. The intersection of any two simplices σ1, σ2 ∈ K is a face of both σ1 and σ2 .

Note that the empty set is a face of every simplex. See also the definition of an abstract simplicial complex, whichloosely speaking is a simplicial complex without an associated geometry.A simplicial k-complex K is a simplicial complex where the largest dimension of any simplex in K equals k. Forinstance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices.A pure or homogeneous simplicial k-complexK is a simplicial complex where every simplex of dimension less thank is a face of some simplex σ ∈ K of dimension exactly k. Informally, a pure 1-complex “looks” like it’s made of abunch of lines, a 2-complex “looks” like it’s made of a bunch of triangles, etc. An example of a non-homogeneouscomplex is a triangle with a line segment attached to one of its vertices.A facet is any simplex in a complex that is not a face of any larger simplex. (Note the difference from a “face” of asimplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension.Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex.For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. Theterm cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definitionof cell complex.The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.

57.2 Closure, star, and link• Two simplices and their closure.

• A vertex and its star.

• A vertex and its link.

Let K be a simplicial complex and let S be a collection of simplices in K.The closure of S (denoted Cl S) is the smallest simplicial subcomplex of K that contains each simplex in S. Cl S isobtained by repeatedly adding to S each face of every simplex in S.The star of S (denoted St S) is the set of all simplices in K that have any faces in S. (Note that the star is generallynot a simplicial complex itself).The link of S (denoted Lk S) equals Cl St S − St Cl S. It is the closed star of S minus the stars of all faces of S.

57.3 Algebraic topology

Main article: Simplicial homology

In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homologygroups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistentorientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces,the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion atpolytope of simplicial complexes as subspaces of Euclidean space, made up of subsets each of which is a simplex. Thatsomewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talkedabout here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topologya compact topological space which is homeomorphic to the geometric realization of a finite simplicial complex isusually called a polyhedron (see Spanier 1966, Maunder 1996, Hilton & Wylie 1967).

57.4. COMBINATORICS 213

57.4 Combinatorics

Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integral sequence (f0, f1, f2, . . . , fd+1), where fi is the number of (i − 1)-dimensional faces of Δ (by convention, f0 = 1 unless Δ is the empty complex).For instance, if Δ is the boundary of the octahedron, then its f-vector is (1, 6, 12, 8), and if Δ is the first simplicialcomplex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors ofsimplicial complexes is given by the Kruskal–Katona theorem.By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial (written in decreasing order ofexponents), we obtain the f-polynomial of Δ. In our two examples above, the f-polynomials would be x3 + 6x2 +12x+ 8 and x4 + 18x3 + 23x2 + 8x+ 1 , respectively.Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coeffi-cients of the polynomial that results from plugging x − 1 into the f-polynomial of Δ. Formally, if we write FΔ(x) tomean the f-polynomial of Δ, then the h-polynomial of Δ is

F∆(x− 1) = h0xd+1 + h1x

d + h2xd−1 + · · ·+ hdx+ hd+1

and the h-vector of Δ is

(h0, h1, h2, · · · , hd+1).

We calculate the h-vector of the octahedron boundary (our first example) as follows:

F (x− 1) = (x− 1)3 + 6(x− 1)2 + 12(x− 1) + 8 = x3 + 3x2 + 3x+ 1.

So the h-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this h-vector is symmetric. Infact, this happens whenever Δ is the boundary of a simplicial polytope (these are the Dehn–Sommerville equations).In general, however, the h-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δto be the 2-complex given by two triangles intersecting only at a common vertex, the resulting h-vector is (1, 3, −2).A complete characterization of all simplicial polytope h-vectors is given by the celebrated g-theorem of Stanley,Billera, and Lee.Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (agraph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch eachother) and as such can be used to determine the combinatorics of sphere packings, such as the number of touchingpairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.

57.5 See also• Abstract simplicial complex• Barycentric subdivision• Causal dynamical triangulation• Delta set• Polygonal chain – 1 dimensional simplicial complex• Tucker’s lemma

57.6 References• Spanier, E.H. (1966), Algebraic Topology, Springer, ISBN 0-387-94426-5• Maunder, C.R.F. (1996), Algebraic Topology, Dover, ISBN 0-486-69131-4• Hilton, P.J.; Wylie, S. (1967), Homology Theory, Cambridge University Press, ISBN 0-521-09422-4

214 CHAPTER 57. SIMPLICIAL COMPLEX

57.7 External links• Weisstein, Eric W., “Simplicial complex”, MathWorld.

Chapter 58

Simplicial group

In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in thecategory of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. Asimplicial group is a Kan complex (in particular, its homotopy groups make sense.) The Dold–Kan correspondencesays that a simplicial abelian group may be identified with a chain complex.A commutative monoid in the category of simplicial abelian groups is a simplicial commutative ring.

58.1 References• Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress in Mathematics 174. Basel, Boston,

Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.

• Charles Weibel, An introduction to homological algebra

58.2 External links• simplicial group in nLab

• http://mathoverflow.net/questions/118500/what-is-a-simplicial-commutative-ring-from-the-point-of-view-of-homotopy-theory/

215

Chapter 59

Simplicial homotopy

In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicialsets. If

f, g : X → Y

are maps between simplicial sets, a simplicial homotopy from f to g is a map

h : X ×1 → Y

such that the obvious diagram (see ) formed by f, g and h commute; the key is to use the diagram that results inf(x) = h(x, 0) and g(x) = h(x, 1) for all x in X.

59.1 See also• Dold–Kan correspondence (under which a chain homotopy corresponds to a simplicial homotopy)

59.2 External links• http://ncatlab.org/nlab/show/simplicial+homotopy

216

Chapter 60

Simplicial manifold

Not to be confused with Symplectic manifold.

In mathematics, the term simplicial manifold commonly refers to either of two different types of objects, whichcombine attributes of a simplex with those of a manifold. Briefly; a simplex is a generalization of the concept ofa triangle into forms with more, or fewer, than two dimensions. Accordingly, a 3-simplex is the figure known as atetrahedron. A manifold is simply a space which appears to be Euclidean (following the laws of ordinary geometry,or more generally a flat Pseudo-Riemannian space) in a given local neighborhood, though it can be greatly morecomplicated overall. The combination of these concepts gives us two useful definitions.

60.1 A manifold made out of simplices

A simplicial manifold is a simplicial complex for which the geometric realization is homeomorphic to a topologicalmanifold. This can mean simply that a neighborhood of each vertex (i.e. the set of simplices that contain that pointas a vertex) is homeomorphic to a n-dimensional ball.A manifold made from simplices can be locally flat, or can approximate a smooth curve, just as a large geodesic domeappears relatively flat over small areas, and approximates a hemisphere over its full extent. One can generalize thisconcept to more dimensions and other kinds of curved surfaces which makes it useful in various kinds of simulations.This notion of simplicial manifold is important in Regge calculus and Causal dynamical triangulations as a way todiscretize spacetime by triangulating it. A simplicial manifold with a metric is called a piecewise linear space.

60.2 A simplicial object built from manifolds

A simplicial manifold is also a simplicial object in the category of manifolds. This is a special case of a simplicialspace in which, for each n, the space of n-simplices is a manifold.For example, if G is a Lie group, then the simplicial nerve of G has the manifold Gn as its space of n-simplices.More generally, G can be a Lie groupoid.

217

Chapter 61

Simplicial presheaf

In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the categoryof topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category ofsimplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. Thenotion was introduced by A. Joyal in the 1970s.[1] Similarly, a simplicial sheaf on a site is a simplicial object in thecategory of sheaves on the site.[2]

Example: Let us consider, say, the étale site of a scheme S. Each U in the site represents the presheaf Hom(−, U) .Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicialsheaf).Example: Let G be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf BG. For example, one might set B GL = lim−→B GLn . These types of examples appear in K-theory.If f : X → Y is a local weak equivalence of simplicial presheaves, then the induced map Zf : ZX → ZY is also alocal weak equivalence.

61.1 Homotopy sheaves of a simplicial presheaf

Let F be a simplicial presheaf on a site. The homotopy sheaves π∗F of F is defined as follows. For any f : X → Yin the site and a 0-simplex s in F(X), set (πpr

0 F )(X) = π0(F (X)) and (πpri (F, s))(f) = πi(F (Y ), f∗(s)) . We

then set πiF to be the sheaf associated with the pre-sheaf πpri F .

61.2 Model structures

The category of simplicial presheaves on a site admits many different model structures.Some of them are obtained by viewing simplicial presheaves as functors

Sop → ∆opSets

The category of such functors is endowed with (at least) three model structures, namely the projective, the Reedy,and the injective model structure. The weak equivalences / fibrations in the first are maps

F → G

such that

F(U)→ G(U)

is a weak equivalence / fibration of simplicial sets, for all U in the site S. The injective model structure is similar, butwith weak equivalences and cofibrations instead.

218

61.3. STACK 219

61.3 Stack

Main article: Stack (mathematics)

A simplicial presheaf F on a site is called a stack if, for any X and any hypercovering H →X, the canonical map

F (X)→ holimF (Hn)

is a weak equivalence as simplicial sets, where the right is the homotopy limit of

[n] = 0, 1, . . . , n 7→ F (Hn)

Any sheaf F on the site can be considered as a stack by viewing F (X) as a constant simplicial set; this way, thecategory of sheaves on the site is included as a subcategory to the homotopy category of simplicial presheaves on thesite. The inclusion functor has a left adjoint and that is exactly F 7→ π0F .If A is a sheaf of abelian group (on the same site), then we define K(A, 1) by doing classifying space constructionlevelwise (the notion comes from the obstruction theory) and set K(A, i) = K(K(A, i− 1), 1) . One can show (byinduction): for any X in the site,

Hi(X;A) = [X,K(A, i)]

where the left denotes a sheaf cohomology and the right the homotopy class of maps.

61.4 See also• cubical set

61.5 Notes[1] http://ncatlab.org/nlab/files/ToenStacksNAC.pdf

[2] Jardine 2007, §1

61.6 Further reading• Konrad Voelkel, Model structures on simplicial presheaves

61.7 References• Jardine, J.F. (2004). “Generalised sheaf cohomology theories”. In Greenlees, J. P. C. Axiomatic, enriched andmotivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, 9-−20 September2002. NATO Science Series II: Mathematics, Physics and Chemistry 131. Dordrecht: Kluwer Academic. pp.29–68. ISBN 1-4020-1833-9. Zbl 1063.55004.

• Jardine, J.F. (2007). “Simplicial presheaves” (PDF).• B. Toën, Simplicial presheaves and derived algebraic geometry

61.8 External links• J.F. Jardine’s homepage

Chapter 62

Simplicial set

In mathematics, a simplicial set is a construction in categorical homotopy theory that is a purely algebraic model ofthe notion of a "well-behaved" topological space. Historically, this model arose from earlier work in combinatorialtopology and in particular from the notion of simplicial complexes. Simplicial sets are used to define quasi-categories,a basic notion of higher category theory.

62.1 Motivation

A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces that can be builtup (or faithfully represented up to homotopy) from simplices and their incidence relations. This is similar to theapproach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purelyalgebraic and do not carry any actual topology (this will become clear in the formal definition).To get back to actual topological spaces, there is a geometric realization functor which turns simplicial sets intocompactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy theory have analogousversions for simplicial sets which generalize these results. While algebraic topologists largely continue to prefer CWcomplexes, there is a growing contingent of researchers interested in using simplicial sets for applications in algebraicgeometry where CW complexes do not naturally exist.

62.2 Intuition

Simplicial sets can be viewed as a higher-dimensional generalization of directed multigraphs. A simplicial set containsvertices (known as “0-simplices” in this context) and arrows (“1-simplices”) between some of these vertices. Twovertices may be connected by several arrows, and directed loops that connect a vertex to itself are also allowed. Unlikedirected multigraphs, simplicial sets may also contain higher simplices. A 2-simplex, for instance, can be thought ofas a two-dimensional “triangular” shape bounded by an ordered list of three vertices A, B, C and three arrows f:A→B,g:B→C and h:A→C. In general, an n-simplex is an object made up from an ordered list of n+1 vertices (which are0-simplices) and n+1 faces (which are (n−1)-simplices). The vertices of the i-th face are the vertices of the n-simplexminus the i-th vertex. The vertices of a simplex need not be distinct and a simplex is not determined by its verticesand faces: two different simplices may share the same list of faces (and therefore the same list of vertices).Simplicial sets should not be confused with abstract simplicial complexes, which generalize simple undirected graphsrather than directed multigraphs.Formally, a simplicial set X is a collection of sets Xn, n=0,1,2,..., together with certain maps between these sets:the face maps dn,i:Xn→Xn−₁ (n=1,2,3,... and 0≤i≤n) and degeneracy maps sn,i:Xn→Xn₊₁ (n=0,1,2,... and 0≤i≤n).We think of the elements of Xn as the n-simplices of X. The map dn,i assigns to each such n-simplex its i-th face,the face “opposite to” (i.e. not containing) the i-th vertex. The map sn,i assigns to each n-simplex the degenerate(n+1)-simplex which arises from the given one by duplicating the i-th vertex. This description implicitly requirescertain consistency relations among the maps dn,i and sn,i. Rather than requiring these simplicial identities explicitlyas part of the definition, the short and elegant modern definition uses the language of category theory.

220

62.3. FORMAL DEFINITION 221

62.3 Formal definition

Let Δ denote the simplex category. The objects of Δ are nonempty linearly ordered sets of the form

[n] = 0, 1, ..., n

with n≥0. The morphisms in Δ are (non-strictly) order-preserving functions between these sets.A simplicial set X is a contravariant functor

X: Δ → Set

where Set is the category of small sets. (Alternatively and equivalently, one may define simplicial sets as covariantfunctors from the opposite category Δop to Set.) Simplicial sets are therefore nothing but presheafs on Δ.Alternatively, one can think of a simplicial set as a simplicial object (see below) in the category Set, but this is onlydifferent language for the definition just given. If we use a covariant functor X: Δ → Set instead of a contravariantone, we arrive at the definition of a cosimplicial set.Simplicial sets form a category, usually denoted sSet, whose objects are simplicial sets and whose morphisms arenatural transformations between them. There is a corresponding category for cosimplicial sets as well, denoted bycSet.

62.4 Face and degeneracy maps

The simplex category Δ is generated by two particularly important families of morphisms (maps), whose imagesunder a given simplicial set functor are called face maps and degeneracy maps of that simplicial set.The face maps of a simplicial set are the images in that simplicial set of the morphisms δ0, . . . , δn : [n− 1]→ [n] ,where δi is the only injection [n−1]→ [n] that “misses” i . Let us denote these face maps by d0, . . . , dn respectively.The degeneracy maps of a simplicial set are the images in that simplicial set of the morphisms σ0, . . . , σn : [n+1]→[n] , where σi is the only surjection [n + 1] → [n] that “hits” i twice. Let us denote these degeneracy maps bys0, . . . , sn respectively.The defined maps satisfy the following simplicial identities:

1. di dj = dj₋₁ di if i < j

2. di sj = sj₋₁ di if i < j

3. di sj = id if i = j or i = j + 1

4. di sj = sj di₋₁ if i > j + 1

5. si sj = sj₊₁ si if i ≤ j.

62.5 Examples

Given a partially ordered set (S,≤), we can define a simplicial set NS, the nerve of S, as follows: for every object [n]of Δ we set NS([n]) = hom ₒ- ₑ ( [n] , S), the order-preserving maps from [n] to S. Every morphism φ:[n]→[m] in Δis an order preserving map, and via composition induces a map NS(φ) : NS([m]) → NS([n]). It is straightforward tocheck that NS is a contravariant functor from Δ to Set: a simplicial set.Concretely, the n-simplices of the nerve NS, i.e. the elements of NSn=NS([n]), can be thought of as ordered length-(n+1) sequences of elements from S: (a0 ≤ a1 ≤ ... ≤ an). The face map di drops the i-th element from such a list,and the degeneracy maps si duplicates the i-th element.A similar construction can be performed for every category C, to obtain the nerve NC of C. Here, NC([n]) is the setof all functors from [n] to C, where we consider [n] as a category with objects 0,1,...,n and a single morphism from ito j whenever i≤j.

222 CHAPTER 62. SIMPLICIAL SET

Concretely, the n-simplices of the nerve NC can be thought of as sequences of n composable morphisms in C:a0→a1→...→an. (In particular, the 0-simplices are the objects of C and the 1-simplices are the morphisms of C.)The face map d0 drops the first morphism from such a list, the face map dn drops the last, and the face map difor 0<i<n composes the (i−1)st and ith morphisms. The degeneracy maps si lengthen the sequence by inserting anidentity morphism at position i.We can recover the poset S from the nerve NS and the category C from the nerve NC; in this sense simplicial setsgeneralize posets and categories.Another important class of examples of simplicial sets is given by the singular set SY of a topological space Y. HereSYn consists of all the continuous maps from the standard topological n-simplex to Y. The singular set is furtherexplained below.

62.6 The standard n-simplex and the category of simplices

The standard n-simplex, denoted Δn, is a simplicial set defined as the functor homΔ(-, [n]) where [n] denotes theordered set 0, 1, ... ,n of the first (n + 1) nonnegative integers. In many texts, it is written instead as hom([n],-)where the homset is understood to be in the opposite category Δop.[1]

The geometric realization |Δn| is just defined to be the standard topological n-simplex in general position given by

|∆n| = (x0, . . . , xn) ∈ Rn+1 : 0 ≤ xi ≤ 1,∑

xi = 1.

By the Yoneda lemma, the n-simplices of a simplicial set X are classified by natural transformations in hom(Δn,X). (Specifically, consider ∆n = ∆op(n,−) , then the Yoneda lemma gives Nat(∆op(n,−), X) ∼= X(n) ) The n-simplices of X are then collectively denoted by Xn. Furthermore, there is a category of simplices, denoted by ∆ ↓ Xwhose objects are maps (i.e. natural transformations) Δn → X and whose morphisms are natural transformations Δn

→ Δm over X arising from maps [n] → [m] in Δ. That is, ∆ ↓ X is a slice category of Δ over X. The followingisomorphism shows that a simplicial set X is a colimit of its simplices:[2]

X ∼= lim−→∆n→X

∆n

where the colimit is taken over the category of simplices of X.

62.7 Geometric realization

There is a functor |•|: sSet→CGHaus called the geometric realization taking a simplicial setX to its correspondingrealization in the category of compactly-generated Hausdorff topological spaces.This larger category is used as the target of the functor because, in particular, a product of simplicial sets

X × Y

is realized as a product

|X| ×Ke |Y |

of the corresponding topological spaces, where ×Ke denotes the Kelley space product. This product is the rightadjoint functor that takes X to XC as described here, applied to the ordinary topological product |X| × |Y |.To define the realization functor, we first define it on n-simplices Δn as the corresponding topological n-simplex |Δn|.The definition then naturally extends to any simplicial set X by setting

|X| = limΔn → X |Δn|

where the colimit is taken over the n-simplex category of X. The geometric realization is functorial on sSet.

62.8. SINGULAR SET FOR A SPACE 223

62.8 Singular set for a space

The singular set of a topological space Y is the simplicial set S(Y) defined by

S(Y)([n]) = homTop(|Δn|, Y) for each object [n] ∈ Δ,

with the obvious functoriality condition on the morphisms. This definition is analogous to a standard idea in singularhomology of “probing” a target topological space with standard topological n-simplices. Furthermore, the singularfunctor S is right adjoint to the geometric realization functor described above, i.e.:

homTₒ (|X|, Y) ≅ homS(X, SY)

for any simplicial set X and any topological space Y.

62.9 Homotopy theory of simplicial sets

In the category of simplicial sets one can define fibrations to be Kan fibrations. A map of simplicial sets is defined tobe a weak equivalence if its geometric realization is a weak equivalence of spaces. A map of simplicial sets is definedto be a cofibration if it is a monomorphism of simplicial sets. It is a difficult theorem of Daniel Quillen that thecategory of simplicial sets with these classes of morphisms satisfies the axioms for a proper closed simplicial modelcategory.A key turning point of the theory is that the geometric realization of a Kan fibration is a Serre fibration of spaces.With the model structure in place, a homotopy theory of simplicial sets can be developed using standard homotopicalalgebra methods. Furthermore, the geometric realization and singular functors give a Quillen equivalence of closedmodel categories inducing an equivalence of homotopy categories

|•|: Ho(sSet) ↔ Ho(Top)

between the homotopy category for simplicial sets and the usual homotopy category of CW complexes with homotopyclasses of maps between them. It is part of the general definition of a Quillen adjunction that the right adjoint functor(in this case, the singular set functor) carries fibrations (resp. trivial fibrations) to fibrations (resp. trivial fibrations).

62.10 Simplicial objects

A simplicial object X in a category C is a contravariant functor

X: Δ → C

or equivalently a covariant functor

X: Δop → C

When C is the category of sets, we are just talking about simplicial sets. Letting C be the category of groups orcategory of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups,respectively.Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlyingsimplicial sets.The homotopy groups of simplicial abelian groups can be computed by making use of the Dold-Kan correspondencewhich yields an equivalence of categories between simplicial abelian groups and bounded chain complexes and isgiven by functors

N: sAb→ Ch+

and

Γ: Ch+ → sAb.

224 CHAPTER 62. SIMPLICIAL SET

62.11 History and uses of simplicial sets

Simplicial sets were originally used to give precise and convenient descriptions of classifying spaces of groups. Thisidea was vastly extended by Grothendieck's idea of considering classifying spaces of categories, and in particular byQuillen's work of algebraic K-theory. In this work, which earned him a Fields Medal, Quillen developed surprisinglyefficient methods for manipulating infinite simplicial sets. Later these methods were used in other areas on the borderbetween algebraic geometry and topology. For instance, the André-Quillen homology of a ring is a “non-abelianhomology”, defined and studied in this way.Both the algebraic K-theory and the André-Quillen homology are defined using algebraic data to write down a sim-plicial set, and then taking the homotopy groups of this simplicial set. Sometimes one simply defines the algebraicK -theory as the space.Simplicial methods are often useful when one wants to prove that a space is a loop space. The basic idea is that ifG is a group with classifying space BG , then G is homotopy equivalent to the loop space ΩBG . If BG itself is agroup, we can iterate the procedure, and G is homotopy equivalent to the double loop space Ω2B(BG) . In case Gis an abelian group, we can actually iterate this infinitely many times, and obtain that G is an infinite loop space.Even if X is not an abelian group, it can happen that it has a composition which is sufficiently commutative so thatone can use the above idea to prove that X is an infinite loop space. In this way, one can prove that the algebraic K-theory of a ring, considered as a topological space, is an infinite loop space.In recent years, simplicial sets have been used in higher category theory and derived algebraic geometry. Quasi-categories can be thought of as categories in which the composition of morphisms is defined only up to homotopy, andinformation about the composition of higher homotopies is also retained. Quasi-categories are defined as simplicialsets satisfying one additional condition, the weak Kan condition.

62.12 See also• Delta set

• Dendroidal set, a generalization of simplicial set.

• Simplicial presheaf

• infinity-category

• Homotopy type theory

• Kan complex

• Dold–Kan correspondence

• Simplicial homotopy

• Simplicial sphere

62.13 Notes[1] S. Gelfand, Yu. Manin, “Methods of Homological Algebra”

[2] Goerss & Jardine, p.7

62.14 References• Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress in Mathematics 174. Basel, Boston,

Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.

• Gelfand, S.; Manin, Yu. Methods of homological algebra.

62.14. REFERENCES 225

• Dylan G.L. Allegretti, Simplicial Sets and van Kampen’s Theorem (An elementary introduction to simplicialsets).

• Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (ed.): Higher K-Theories. Lecture Notes in Mathe-matics, vol. 341. Springer-Verlag, Berlin 1973. ISBN 3-540-06434-6

• G.B. Segal, Categories and cohomology theories, Topology, 13, (1974), 293 - 312.

• simplicial set in nLab

Chapter 63

Sperner’s theorem

Sperner’s theorem, in discrete mathematics, describes the largest possible families of finite sets none of whichcontain any other sets in the family. It is one of the central results in extremal set theory, and is named after EmanuelSperner, who published it in 1928.This result is sometimes called Sperner’s lemma, but the name "Sperner’s lemma" also refers to an unrelated resulton coloring triangulations. To differentiate the two results, the result on the size of a Sperner family is now morecommonly known as Sperner’s theorem.

63.1 Statement

A family of sets that does not include two sets X and Y for which X ⊂ Y is called a Sperner family. For example,the family of k-element subsets of an n-element set is a Sperner family. No set in this family can contain any of theothers, because a containing set has to be strictly bigger than the set it contains, and in this family all sets have equalsize. The value of k that makes this example have as many sets as possible is n/2 if n is even, or the nearest integerto n/2 if n is odd. For this choice, the number of sets in the family is

(n

⌊n/2⌋)

.Sperner’s theorem states that these examples are the largest possible Sperner families over an n-element set. Formally,the theorem states that, for every Sperner family S whose union has a total of n elements,

|S| ≤(

n

⌊n/2⌋

).

63.2 Partial orders

Sperner’s theorem can also be stated in terms of partial order width. The family of all subsets of an n-element set(its power set) can be partially ordered by set inclusion; in this partial order, two distinct elements are said to beincomparable when neither of them contains the other. The width of a partial order is the largest number of elementsin an antichain, a set of pairwise incomparable elements. Translating this terminology into the language of sets, anantichain is just a Sperner family, and the width of the partial order is the maximum number of sets in a Spernerfamily. Thus, another way of stating Sperner’s theorem is that the width of the inclusion order on a power set is(

n⌊n/2⌋

).

A graded partially ordered set is said to have the Sperner property when one of its largest antichains is formed by aset of elements that all have the same rank. In this terminology, Sperner’s theorem states that the partially orderedset of all subsets of a finite set, partially ordered by set inclusion, has the Sperner property.

63.3 Proof

The following proof is due to Lubell (1966). Let sk denote the number of k-sets in S. For all 0 ≤ k ≤ n,

226

63.4. GENERALIZATIONS 227

(n

⌊n/2⌋

)≥(n

k

)and, thus,

sk(n

⌊n/2⌋) ≤ sk(

nk

) .Since S is an antichain, we can sum over the above inequality from k = 0 to n and then apply the LYM inequality toobtain

n∑k=0

sk(n

⌊n/2⌋) ≤ n∑

k=0

sk(nk

) ≤ 1,

which means

|S| =n∑

k=0

sk ≤(

n

⌊n/2⌋

).

This completes the proof.

63.4 Generalizations

There are several generalizations of Sperner’s theorem for subsets of P(E), the poset of all subsets of E.

63.4.1 No long chains

A chain is a subfamily S0, S1, . . . , Sr ⊆ P(E) that is totally ordered, i.e., S0 ⊂ S1 ⊂ · · · ⊂ Sr (possibly afterrenumbering). The chain has r + 1 members and length r. An r-chain-free family (also called an r-family) is afamily of subsets of E that contains no chain of length r. Erdős (1945) proved that the largest size of an r-chain-freefamily is the sum of the r largest binomial coefficients

(ni

). The case r = 1 is Sperner’s theorem.

63.4.2 p-compositions of a set

In the setP(E)p of p-tuples of subsets of E, we say a p-tuple (S1, . . . , Sp) is ≤ another one, (T1, . . . , Tp), if Si ⊆ Ti

for each i = 1,2,...,p. We call (S1, . . . , Sp) a p-composition of E if the sets S1, . . . , Sp form a partition of E.Meshalkin (1963) proved that the maximum size of an antichain of p-compositions is the largest p-multinomialcoefficient

(n

n1 n2 ... np

), that is, the coefficient in which all ni are as nearly equal as possible (i.e., they differ by at

most 1). Meshalkin proved this by proving a generalized LYM inequality.The case p = 2 is Sperner’s theorem, because then S2 = E \ S1 and the assumptions reduce to the sets S1 being aSperner family.

63.4.3 No long chains in p-compositions of a set

Beck & Zaslavsky (2002) combined the Erdös and Meshalkin theorems by adapting Meshalkin’s proof of his gener-alized LYM inequality. They showed that the largest size of a family of p-compositions such that the sets in the i-thposition of the p-tuples, ignoring duplications, are r-chain-free, for every i = 1, 2, . . . , p− 1 (but not necessarily fori = p), is not greater than the sum of the rp−1 largest p-multinomial coefficients.

228 CHAPTER 63. SPERNER’S THEOREM

63.4.4 Projective geometry analog

In the finite projective geometry PG(d, Fq) of dimension d over a finite field of order q, let L(p, Fq) be the familyof all subspaces. When partially ordered by set inclusion, this family is a lattice. Rota & Harper (1971) proved thatthe largest size of an antichain in L(p, Fq) is the largest Gaussian coefficient

[d+ 1k

]; this is the projective-geometry

analog, or q-analog, of Sperner’s theorem.They further proved that the largest size of an r-chain-free family in L(p, Fq) is the sum of the r largest Gaussiancoefficients. Their proof is by a projective analog of the LYM inequality.

63.4.5 No long chains in p-compositions of a projective space

Beck & Zaslavsky (2003) obtained a Meshalkin-like generalization of the Rota–Harper theorem. In PG(d, Fq), aMeshalkin sequence of length p is a sequence (A1, . . . , Ap) of projective subspaces such that no proper subspaceof PG(d, Fq) contains them all and their dimensions sum to d − p + 1 . The theorem is that a family of Meshalkinsequences of length p in PG(d, Fq), such that the subspaces appearing in position i of the sequences contain no chainof length r for each i = 1, 2, . . . , p− 1, is not more than the sum of the largest rp−1 of the quantities

[d+ 1

n1 n2 . . . np

]qs2(n1,...,np),

where[

d+ 1n1 n2 . . . np

](in which we assume that d+ 1 = n1 + · · ·+ np ) denotes the p-Gaussian coefficient

[d+ 1n1

][d+ 1− n1

n2

]· · ·[d+ 1− (n1 + · · ·+ np−1)

np

]and

s2(n1, . . . , np) := n1n2 + n1n3 + n2n3 + n1n4 + · · ·+ np−1np,

the second elementary symmetric function of the numbers n1, n2, . . . , np.

63.5 References

• Anderson, Ian (1987), “Sperner’s theorem”, Combinatorics of Finite Sets, Oxford University Press, pp. 2–4.

• Beck, Matthias; Zaslavsky, Thomas (2002), “A shorter, simpler, stronger proof of the Meshalkin-Hochberg-Hirsch bounds on componentwise antichains”, Journal of Combinatorial Theory, Series A 100 (1): 196–199,doi:10.1006/jcta.2002.3295, MR 1932078.

• Beck, Matthias; Zaslavsky, Thomas (2003), “A Meshalkin theorem for projective geometries”, Journal ofCombinatorial Theory, Series A 102 (2): 433–441, doi:10.1016/S0097-3165(03)00049-9, MR 1979545.

• Engel, Konrad (1997), Sperner theory, Encyclopedia of Mathematics and its Applications 65, Cambridge:Cambridge University Press, p. x+417, doi:10.1017/CBO9780511574719, ISBN 0-521-45206-6, MR 1429390.

• Engel, K. (2001), “Sperner theorem”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Erdős, P. (1945), “On a lemma of Littlewood and Offord” (PDF),Bulletin of the AmericanMathematical Society51: 898–902, doi:10.1090/S0002-9904-1945-08454-7, MR 0014608

• Lubell, D. (1966), “A short proof of Sperner’s lemma”, Journal of Combinatorial Theory 1 (2): 299, doi:10.1016/S0021-9800(66)80035-2, MR 0194348.

63.6. EXTERNAL LINKS 229

• Meshalkin, L.D. (1963), “Generalization of Sperner’s theorem on the number of subsets of a finite set. (InRussian)", Theory of Probability and its Applications 8 (2): 203–204, doi:10.1137/1108023.

• Rota, Gian-Carlo; Harper, L. H. (1971), “Matching theory, an introduction”, Advances in Probability andRelated Topics, Vol. 1, New York: Dekker, pp. 169–215, MR 0282855.

• Sperner, Emanuel (1928), “Ein Satz über Untermengen einer endlichen Menge”, Mathematische Zeitschrift (inGerman) 27 (1): 544–548, doi:10.1007/BF01171114, JFM 54.0090.06.

63.6 External links• Sperner’s Theorem at cut-the-knot

• Sperner’s theorem on the polymath1 wiki

Chapter 64

Steiner system

The Fano plane is an S(2,3,7) Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongsto a unique line.

In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically at-design with λ = 1 and t ≥ 2.

230

64.1. EXAMPLES 231

A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsetsof S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In an alternatenotation for block designs, an S(t,k,n) would be a t-(n,k,1) design.This definition is relatively modern, generalizing the classical definition of Steiner systems which in addition requiredthat k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple (or triad) system, while an S(3,4,n) was called aSteiner quadruple system, and so on. With the generalization of the definition, this naming system is no longer strictlyadhered to.A long-standing problem in design theory is if any nontrivial (t < k < n) Steiner systems have t ≥ 6; also if infinitelymany have t = 4 or 5.[1] This was claimed to be solved in the affirmative by Peter Keevash.[2][3]

64.1 Examples

64.1.1 Finite projective planes

A finite projective plane of order q, with the lines as blocks, is an S(2, q + 1, q2 + q + 1) , since it has q2 + q + 1points, each line passes through q + 1 points, and each pair of distinct points lies on exactly one line.

64.1.2 Finite affine planes

A finite affine plane of order q, with the lines as blocks, is an S(2, q, q2). An affine plane of order q can be obtainedfrom a projective plane of the same order by removing one block and all of the points in that block from the projectiveplane. Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.

64.2 Classical Steiner systems

64.2.1 Steiner triple systems

An S(2,3,n) is called a Steiner triple system, and its blocks are called triples. It is common to see the abbreviationSTS(n) for a Steiner triple system of order n. The number of triples is n(n−1)/6. A necessary and sufficient conditionfor the existence of an S(2,3,n) is that n ≡ 1 or 3 (mod 6). The projective plane of order 2 (the Fano plane) is anSTS(7) and the affine plane of order 3 is an STS(9).Up to isomorphism, the STS(7) and STS(9) are unique, there are two STS(13)s, 80 STS(15)s, and 11,084,874,829STS(19)s.[4]

We can define a multiplication on the set S using the Steiner triple system by setting aa = a for all a in S, and ab= c if a,b,c is a triple. This makes S an idempotent, commutative quasigroup. It has the additional property that“ab” = “c” implies “bc” = “a” and “ca” = “b”.[5] Conversely, any (finite) quasigroup with these properties arises froma Steiner triple system. Commutative idempotent quasigroups satisfying this additional property are called Steinerquasigroups.[6]

64.2.2 Steiner quadruple systems

An S(3,4,n) is called a Steiner quadruple system. A necessary and sufficient condition for the existence of anS(3,4,n) is that n ≡ 2 or 4 (mod 6). The abbreviation SQS(n) is often used for these systems.Up to isomorphism, SQS(8) and SQS(10) are unique, there are 4 SQS(14)s and 1,054,163 SQS(16)s.[7]

64.2.3 Steiner quintuple systems

An S(4,5,n) is called a Steiner quintuple system. A necessary condition for the existence of such a system is that n≡ 3or 5 (mod 6) which comes from considerations that apply to all the classical Steiner systems. An additional necessarycondition is that n ≡ 4 (mod 5), which comes from the fact that the number of blocks must be an integer. Sufficientconditions are not known.

232 CHAPTER 64. STEINER SYSTEM

There is a unique Steiner quintuple system of order 11, but none of order 15 or order 17.[8] Systems are known fororders 23, 35, 47, 71, 83, 107, 131, 167 and 243. The smallest order for which the existence is not known (as of2011) is 21.

64.3 Properties

It is clear from the definition of S(t,k,n) that 1 < t < k < n . (Equalities, while technically possible, lead to trivialsystems.)If S(t,k,n) exists, then taking all blocks containing a specific element and discarding that element gives a derived systemS(t−1,k−1,n−1). Therefore the existence of S(t−1,k−1,n−1) is a necessary condition for the existence of S(t,k,n).The number of t-element subsets in S is

(nt

), while the number of t-element subsets in each block is

(kt

). Since

every t-element subset is contained in exactly one block, we have(nt

)= b(kt

), or b =

(nt

)(kt

) , where b is the number

of blocks. Similar reasoning about t-element subsets containing a particular element gives us(n−1t−1

)= r

(k−1t−1

),

or r =

(n−1t−1

)(k−1t−1

) , where r is the number of blocks containing any given element. From these definitions follows the

equation bk = rn . It is a necessary condition for the existence of S(t,k,n) that b and r are integers. As with anyblock design, Fisher’s inequality b ≥ n is true in Steiner systems.Given the parameters of a Steiner system S(t,k,n) and a subset of size t′ ≤ t , contained in at least one block, onecan compute the number of blocks intersecting that subset in a fixed number of elements by constructing a Pascaltriangle.[9] In particular, the number of blocks intersecting a fixed block in any number of elements is independent ofthe chosen block.It can be shown that if there is a Steiner system S(2,k,n), where k is a prime power greater than 1, then n ≡ 1 or k(mod k(k−1)). In particular, a Steiner triple system S(2,3,n) must have n = 6m+1 or 6m+3. It is known that this is theonly restriction on Steiner triple systems, that is, for each natural number m, systems S(2,3,6m+1) and S(2,3,6m+3)exist.

64.4 History

Steiner triple systems were defined for the first time by W.S.B. Woolhouse in 1844 in the Prize question #1733 ofLady’s and Gentlemen’s Diary.[10] The posed problem was solved by Thomas Kirkman (1847). In 1850 Kirkmanposed a variation of the problem known as Kirkman’s schoolgirl problem, which asks for triple systems having anadditional property (resolvability). Unaware of Kirkman’s work, Jakob Steiner (1853) reintroduced triple systems,and as this work was more widely known, the systems were named in his honor.

64.5 Mathieu groups

Several examples of Steiner systems are closely related to group theory. In particular, the finite simple groups calledMathieu groups arise as automorphism groups of Steiner systems:

• The Mathieu group M11 is the automorphism group of a S(4,5,11) Steiner system

• The Mathieu group M12 is the automorphism group of a S(5,6,12) Steiner system

• The Mathieu group M22 is the unique index 2 subgroup of the automorphism group of a S(3,6,22) Steinersystem

• The Mathieu group M23 is the automorphism group of a S(4,7,23) Steiner system

• The Mathieu group M24 is the automorphism group of a S(5,8,24) Steiner system.

64.6. THE STEINER SYSTEM S(5, 6, 12) 233

64.6 The Steiner system S(5, 6, 12)

There is a unique S(5,6,12) Steiner system; its automorphism group is the Mathieu group M12, and in that context itis denoted by W12.

64.6.1 Constructions

There are different ways to construct an S(5,6,12) system.

Projective line method

This construction is due to Carmichael (1937).[11]

Add a new element, call it ∞, to the 11 elements of the finite field F11 (that is, the integers mod 11). This set, S,of 12 elements can be formally identified with the points of the projective line over F11. Call the following specificsubset of size 6,

∞, 1, 3, 4, 5, 9,

a “block”. From this block, we obtain the other blocks of the S(5,6,12) system by repeatedly applying the linearfractional transformations:

z′ = f(z) =az + b

cz + dwhere ,a, b, c, d in are F11 and ad− bc in square non-zero a is F11.

With the usual conventions of defining f (−d/c) = ∞ and f (∞) = a/c, these functions map the set S onto itself. Ingeometric language, they are projectivities of the projective line. They form a group under composition which is theprojective special linear group PSL(2,11) of order 660. There are exactly five elements of this group that leave thestarting block fixed setwise,[12] so there will be 132 images of that block. As a consequence of the multiply transitiveproperty of this group acting on this set, any subset of five elements of S will appear in exactly one of these 132images of size six.

Kitten method

An alternative construction of W12 is obtained by use of the 'kitten' of R.T. Curtis,[13] which was intended as a “handcalculator” to write down blocks one at a time. The kitten method is based on completing patterns in a 3x3 grid ofnumbers, which represent an affine geometry on the vector space F3xF3, an S(2,3,9) system.

Construction from K6 graph factorization

The relations between the graph factors of the complete graph K6 generate an S(5,6,12).[14] A K6 graph has 6 different1-factorizations (ways to partition the edges into disjoint perfect matchings), and also 6 vertices. The set of verticesand the set of factorizations provide one block each. For every distinct pair of factorizations, there exists exactly oneperfect matching in common. Take the set of vertices and replace the two vertices corresponding to an edge of thecommon perfect matching with the labels corresponding to the factorizations; add that to the set of blocks. Repeatthis with the other two edges of the common perfect matching. Similarly take the set of factorizations and replacethe labels corresponding to the two factorizations with the end points of an edge in the common perfect matching.Repeat with the other two edges in the matching. There are thus 3+3 = 6 blocks per pair of factorizations, and thereare 6C2 = 15 pairs among the 6 factorizations, resulting in 90 new blocks. Finally take the full set of 12C6 = 924combinations of 6 objects out of 12, and discard any combination that has 5 or more objects in common with any ofthe 92 blocks generated so far. Exactly 40 blocks remain, resulting in 2+90+40 = 132 blocks of the S(5,6,12).

234 CHAPTER 64. STEINER SYSTEM

64.7 The Steiner system S(5, 8, 24)

The Steiner system S(5, 8, 24), also known as the Witt design or Witt geometry, was first described by Carmichael(1931) and rediscovered by Witt (1938). This system is connected with many of the sporadic simple groups and withthe exceptional 24-dimensional lattice known as the Leech lattice.The automorphism group of S(5, 8, 24) is the Mathieu group M24, and in that context the design is denoted W24

(“W” for “Witt”)

64.7.1 Constructions

There are many ways to construct the S(5,8,24). Two methods are described here:

Method based on 8-combinations of 24 elements

All 8-element subsets of a 24-element set are generated in lexicographic order, and any such subset which differsfrom some subset already found in fewer than four positions is discarded.The list of octads for the elements 01, 02, 03, ..., 22, 23, 24 is then:

01 02 03 04 05 06 07 0801 02 03 04 09 10 11 1201 02 03 04 13 14 15 16.. (next 753 octads omitted).13 14 15 16 17 18 19 2013 14 15 16 21 22 23 2417 18 19 20 21 22 23 24

Each single element occurs 253 times somewhere in some octad. Each pair occurs 77 times. Each triple occurs 21times. Each quadruple (tetrad) occurs 5 times. Each quintuple (pentad) occurs once. Not every hexad, heptad oroctad occurs.

Method based on 24-bit binary strings

All 24-bit binary strings are generated in lexicographic order, and any such string that differs from some earlier onein fewer than 8 positions is discarded. The result looks like this:000000000000000000000000 000000000000000011111111 000000000000111100001111 000000000000111111110000000000000011001100110011 000000000011001111001100 000000000011110000111100 000000000011110011000011000000000101010101010101 000000000101010110101010 . . (next 4083 24-bit strings omitted) . 111111111111000011110000111111111111111100000000 111111111111111111111111The list contains 4096 items, which are each code words of the extended binary Golay code. They form a groupunder the XOR operation. One of them has zero 1-bits, 759 of them have eight 1-bits, 2576 of them have twelve1-bits, 759 of them have sixteen 1-bits, and one has twenty-four 1-bits. The 759 8-element blocks of the S(5,8,24)(called octads) are given by the patterns of 1’s in the code words with eight 1-bits.

64.8 See also• Constant weight code

• Kirkman’s schoolgirl problem

• Sylvester–Gallai configuration

64.9. NOTES 235

64.9 Notes[1] “Encyclopaedia of Design Theory: t-Designs”. Designtheory.org. 2004-10-04. Retrieved 2012-08-17.

[2] Keevash, Peter (2014). “The existence of designs”. arXiv:1401.3665.

[3] “A Design Dilemma Solved, Minus Designs”. Quanta Magazine. 2015-06-09. Retrieved 2015-06-27.

[4] Colbourn & Dinitz 2007, pg.60

[5] This property is equivalent to saying that (xy)y = x for all x and y in the idempotent commutative quasigroup.

[6] Colbourn & Dinitz 2007, pg. 497, definition 28.12

[7] Colbourn & Dinitz 2007, pg.106

[8] Östergard & Pottonen 2008

[9] Assmus & Key 1994, pg. 8

[10] Lindner & Rodger 1997, pg.3

[11] Carmichael 1956, p. 431

[12] Beth, Jungnickel & Lenz 1986, p. 196

[13] Curtis 1984

[14] EAGTS textbook

64.10 References• Assmus, E. F., Jr.; Key, J. D. (1994), “8. Steiner Systems”, Designs and Their Codes, Cambridge University

Press, pp. 295–316, ISBN 0-521-45839-0.• Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge University

Press. 2nd ed. (1999) ISBN 978-0-521-44432-3.• Carmichael, Robert (1931), “Tactical Configurations of Rank Two”, American Journal of Mathematics 53:

217–240, doi:10.2307/2370885• Carmichael, Robert D. (1956) [1937], Introduction to the theory of Groups of Finite Order, Dover, ISBN 0-

486-60300-8• Colbourn, Charles J.; Dinitz, Jeffrey H. (1996), Handbook of Combinatorial Designs, Boca Raton: Chapman

& Hall/ CRC, ISBN 0-8493-8948-8, Zbl 0836.00010• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:

Chapman & Hall/ CRC, ISBN 1-58488-506-8, Zbl 1101.05001• Curtis, R.T. (1984), “The Steiner system S(5,6,12), the Mathieu group M12 and the “kitten"", in Atkinson,

Michael D., Computational group theory (Durham, 1982), London: Academic Press, pp. 353–358, ISBN0-12-066270-1, MR 0760669

• Hughes, D. R.; Piper, F. C. (1985), Design Theory, Cambridge University Press, pp. 173–176, ISBN 0-521-35872-8.

• Kirkman, Thomas P. (1847), “On a Problem in Combinations”, The Cambridge and Dublin MathematicalJournal (Macmillan, Barclay, and Macmillan) II: 191–204.

• Lindner, C.C.; Rodger, C.A. (1997), Design Theory, Boca Raton: CRC Press, ISBN 0-8493-3986-3• Östergard, Patric R.J.; Pottonen, Olli (2008), “There exists no Steiner system S(4,5,17)", Journal of Combina-torial Theory Series A 115 (8): 1570–1573, doi:10.1016/j.jcta.2008.04.005

• Steiner, J. (1853), “Combinatorische Aufgabe”, Journal für die Reine und Angewandte Mathematik 45: 181–182.

• Witt, Ernst (1938), “Die 5-Fach transitiven Gruppen von Mathieu”, Abh. Math. Sem. Univ. Hamburg 12:256–264, doi:10.1007/BF02948947

236 CHAPTER 64. STEINER SYSTEM

64.11 External links• Rowland, Todd and Weisstein, Eric W., “Steiner System”, MathWorld.

• Rumov, B.T. (2001), “Steiner system”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Steiner systems by Andries E. Brouwer

• Implementation of S(5,8,24) by Dr. Alberto Delgado, Gabe Hart, and Michael Kolkebeck

• S(5, 8, 24) Software and Listing by Johan E. Mebius

• The Witt Design computed by Ashay Dharwadker

Chapter 65

Symmetric spectrum

In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an actionof the symmetric group Σn on Xn such that the composition of structure maps

S1 ∧ · · · ∧ S1 ∧Xn → S1 ∧ · · · ∧ S1 ∧Xn+1 → · · · → S1 ∧Xn+p−1 → Xn+p

is equivariant with respect to Σp × Σn . A morphism between symmetric spectra is a morphism of spectra that isequivariant with respect to the actions of symmetric groups.The technical advantage of the category SpΣ of symmetric spectra is that it has a closed symmetric monoidal structure(with respect to smash product). It is also a simplicial model category. A symmetric ring spectrum is a monoidin SpΣ ; if the monoid is commutative, it’s a commutative ring spectrum. The possibility of this definition of “ringspectrum” was one of motivations behind the category.A similar technical goal is also achieved by May’s theory of S-modules, a competing theory.

65.1 References• Introduction to symmetric spectra I

• M. Hovey, B. Shipley, and J. Smith, “Symmetric spectra”, Journal of the AMS 13 (1999), no. 1, 149 – 208.

237

Chapter 66

Teichmüller–Tukey lemma

In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey’s lemma,), named after John Tukeyand Oswald Teichmüller, states that every nonempty collection of finite character has a maximal element with respectto inclusion. It is equivalent to the Axiom of Choice.

66.1 Definitions

A family of sets is of finite character provided it has the following properties:

1. For each A ∈ F , every finite subset of A belongs to F .

2. If every finite subset of a given set A belongs to F , then A belongs to F .

66.2 Applications

In linear algebra, the lemma may be used to show the existence of a basis. Let V be a vector space. Consider thecollectionF of linearly independent sets of vectors. This is a collection of finite character Thus, a maximal set exists,which must then span V and be a basis for V.

66.3 See also• Well-ordering theorem

• Hausdorff maximal principle

66.4 References• Brillinger, David R. “John Wilder Tukey”

238

Chapter 67

Tverberg’s theorem

A Tverberg partition of the vertices of a regular heptagon into three subsets with intersecting convex hulls.

In discrete geometry, Tverberg’s theorem, first stated by Helge Tverberg (1966), is the result that sufficiently manypoints in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically,for any set of

239

240 CHAPTER 67. TVERBERG’S THEOREM

(d+ 1)(r − 1) + 1

points there exists a point x (not necessarily one of the given points) and a partition of the given points into r subsets,such that x belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as aTverberg partition.

67.1 Examples

For r = 2, Tverberg’s theorem states that any d + 2 points may be partitioned into two subsets with intersecting convexhulls; this special case is known as Radon’s theorem. In this case, for points in general position, there is a uniquepartition.The case r = 3 and d = 2 states that any seven points in the plane may be partitioned into three subsets with intersectingconvex hulls. The illustration shows an example in which the seven points are the vertices of a regular heptagon. Asthe example shows, there may be many different Tverberg partitions of the same set of points; these seven points maybe partitioned in seven different ways that differ by rotations of each other.

67.2 See also• Rota’s basis conjecture

67.3 References• Tverberg, H. (1966), “A generalization of Radon’s theorem” (PDF), Journal of the London MathematicalSociety 41: 123–128, doi:10.1112/jlms/s1-41.1.123.

• Hell, S. (2006), Tverberg-type theorems and the Fractional Helly property, Dissertation, TU Berlin.

Chapter 68

Two-graph

In mathematics, a two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (un-ordered) quadruple from X contains an even number of triples of the two-graph. A regular two-graph has theproperty that every pair of vertices lies in the same number of triples of the two-graph. Two-graphs have been stud-ied because of their connection with equiangular lines and, for regular two-graphs, strongly regular graphs, and alsofinite groups because many regular two-graphs have interesting automorphism groups.A two-graph is not a graph and should not be confused with other objects called 2-graphs in graph theory, such as2-regular graphs.

68.1 Examples

On the set of vertices 1,...,6 the following collection of unordered triples is a two-graph:

123 124 135 146 156 236 245 256 345 346

This two-graph is a regular two-graph since each pair of distinct vertices appears together in exactly two triples.Given a simple graph G = (V,E), the set of triples of the vertex set V whose induced subgraph has an odd number ofedges forms a two-graph on the set V. Every two-graph can be represented in this way.[1] This example is referred toas the standard construction of a two-graph from a simple graph.As a more complex example, let T be a tree with edge set E. The set of all triples of E that are not contained in apath of T form a two-graph on the set E.[2]

68.2 Switching and graphs

A two-graph is equivalent to a switching class of graphs and also to a (signed) switching class of signed completegraphs.Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in theset and the other not in the set: thus the edge set is changed so that an adjacent pair becomes nonadjacent and anonadjacent pair becomes adjacent. The edges whose endpoints are both in the set, or both not in the set, are notchanged. Graphs are switching equivalent if one can be obtained from the other by switching. An equivalence classof graphs under switching is called a switching class. Switching was introduced by van Lint & Seidel (1966) anddeveloped by Seidel; it has been called graph switching or Seidel switching, partly to distinguish it from switchingof signed graphs.In the standard construction of a two-graph from a simple graph given above, two graphs will yield the same two-graphif and only if they are equivalent under switching, that is, they are in the same switching class.Let Γ be a two-graph on the set X. For any element x of X, define a graph with vertex set X having vertices y and zadjacent if and only if x, y, z is in Γ. In this graph, x will be an isolated vertex. This construction is reversible; given

241

242 CHAPTER 68. TWO-GRAPH

X X

Y Y

Switching X,Y in a graph

a simple graph G, adjoin a new element x to the set of vertices of G and define the two-graph whose triples are all thex, y, z where y and z are adjacent vertices in G. This two-graph is called the extension of G by x in design theoreticlanguage.[3] In a given switching class of graphs of a regular two-graph, let Γx be the unique graph having x as anisolated vertex (this always exists, just take any graph in the class and switch the open neighborhood of x) withoutthe vertex x. That is, the two-graph is the extension of Γx by x. In the first example above of a regular two-graph, Γxis a 5-cycle for any choice of x.[4]

To a graph G there corresponds a signed complete graph Σ on the same vertex set, whose edges are signed negativeif in G and positive if not in G. Conversely, G is the subgraph of Σ that consists of all vertices and all negative edges.The two-graph of G can also be defined as the set of triples of vertices that support a negative triangle (a triangle withan odd number of negative edges) in Σ. Two signed complete graphs yield the same two-graph if and only if they areequivalent under switching.Switching ofG and of Σ are related: switching the same vertices in both yields a graphH and its corresponding signedcomplete graph.

68.3. ADJACENCY MATRIX 243

68.3 Adjacency matrix

The adjacency matrix of a two-graph is the adjacency matrix of the corresponding signed complete graph; thus it issymmetric, is zero on the diagonal, and has entries ±1 off the diagonal. If G is the graph corresponding to the signedcomplete graph Σ, this matrix is called the (0, −1, 1)-adjacency matrix or Seidel adjacency matrix of G. The Seidelmatrix has zero entries on the main diagonal,−1 entries for adjacent vertices and +1 entries for non-adjacent vertices.If graphs G and H are in a same switching class, the multisets of eigenvalues of the two Seidel adjacency matrices ofG and H coincide, since the matrices are similar.[5]

A two-graph on a set V is regular if and only if its adjacency matrix has just two distinct eigenvalues ρ1 > 0 > ρ2 say,where ρ1ρ2 = 1 - |V |.[6]

68.4 Equiangular lines

Main article: Equiangular lines

Every two-graph is equivalent to a set of lines in some dimensional euclidean space each pair of which meet in thesame angle. The set of lines constructed from a two graph on n vertices is obtained as follows. Let -ρ be the smallesteigenvalue of the Seidel adjacency matrix, A, of the two-graph, and suppose that it has multiplicity n - d. Thenthe matrix ρI + A is positive semi-definite of rank d and thus can be represented as the Gram matrix of the innerproducts of n vectors in euclidean d-space. As these vectors have the same norm (namely, √ρ ) and mutual innerproducts ±1, any pair of the n lines spanned by them meet in the same angle φ where cos φ = 1/ρ. Conversely, anyset of non-orthogonal equiangular lines in a euclidean space can give rise to a two-graph (see equiangular lines forthe construction).[7]

With the notation as above, the maximum cardinality n satisfies n ≤ d(ρ2 - 1)/(ρ2 - d) and the bound is achieved ifand only if the two-graph is regular.

68.5 Strongly regular graphs

Main article: Strongly regular graph

The two-graphs on X consisting of all possible triples of X and no triples of X are regular two-graphs and are con-sidered to be trivial two-graphs.For non-trivial two-graphs on the set X, the two-graph is regular if and only if for some x in X the graph Γx is astrongly regular graph with k = 2μ (the degree of any vertex is twice the number of vertices adjacent to both of anynon-adjacent pair of vertices). If this condition holds for one x in X, it holds for all the elements of X.[8]

It follows that a non-trivial regular two-graph has an even number of points.If G is a regular graph whose two-graph extension is Γ having n points, then Γ is a regular two-graph if and only if Gis a strongly regular graph with eigenvalues k, r and s satisfying n = 2(k - r) or n = 2(k - s).[9]

68.6 Notes[1] Colburn & Dinitz 2007, p. 876, Remark 13.2

[2] Cameron, P.J. (1994), “Two-graphs and trees”, Discrete Mathematics 127: 63–74, doi:10.1016/0012-365x(92)00468-7cited in Colburn & Dinitz 2007, p. 876, Construction 13.12

[3] Cameron & van Lint 1991, pp. 58-59

[4] Cameron & van Lint 1991, p. 62

[5] Cameron & van Lint 1991, p. 61

[6] Colburn & Dinitz 2007, p. 878 #13.24

244 CHAPTER 68. TWO-GRAPH

[7] van Lint & Seidel 1966

[8] Cameron & van Lint 1991, p. 62 Theorem 4.11

[9] Cameron & van Lint 1991, p. 62 Theorem 4.12

68.7 References• Brouwer, A.E., Cohen, A.M., and Neumaier, A. (1989), Distance-Regular Graphs. Springer-Verlag, Berlin.

Sections 1.5, 3.8, 7.6C.

• Cameron, P.J.; van Lint, J.H. (1991), Designs, Graphs, Codes and their Links, London Mathematical SocietyStudent Texts 22, Cambridge University Press, ISBN 978-0-521-42385-4

• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, pp. 875–882, ISBN 1-58488-506-8

• Godsil, Chris: Royle, Gordon (2001), Algebraic Graph Theory. Graduate Texts in Mathematics, Vol. 207.Springer-Verlag, New York. Chapter 11.

• Seidel, J. J. (1976), A survey of two-graphs. In: Colloquio Internazionale sulle Teorie Combinatorie (Proceed-ings, Rome, 1973), Vol. I, pp. 481–511. Atti dei Convegni Lincei, No. 17. Accademia Nazionale dei Lincei,Rome.

• Taylor, D. E. (1977), Regular 2-graphs. Proceedings of the London Mathematical Society (3), vol. 35, pp.257–274.

• van Lint, J. H.; Seidel, J. J. (1966), “Equilateral point sets in elliptic geometry”, Indagationes Mathematicae,Proc. Koninkl. Ned. Akad. Wetenschap. Ser. A 69 28: 335–348

Chapter 69

Ultrafilter

In the mathematical field of set theory, an ultrafilter is a maximal filter, that is, a filter that cannot be enlarged. Filtersand ultrafilters are special subsets of partially ordered sets. Ultrafilters can also be defined on Boolean algebras andsets:

• An ultrafilter on a poset P is a maximal filter on P.

• An ultrafilter on a Boolean algebra B is an ultrafilter on the poset of non-zero elements of B.

• An ultrafilter on a set X is an ultrafilter on the Boolean algebra of subsets of X.

Rather confusingly, an ultrafilter on a poset P or Boolean algebra B is a subset of P or B, while an ultrafilter on a setX is a collection of subsets of X. Ultrafilters have many applications in set theory, model theory, and topology.An ultrafilter on a set X has some special properties. For example, given any subset A of X, the ultrafilter mustcontain either A or its complement X \ A. In addition, an ultrafilter on a set X may be considered as a finitely additivemeasure. In this view, every subset of X is either considered "almost everything" (has measure 1) or “almost nothing”(has measure 0).

69.1 Formal definition for ultrafilter on a set

Given a set X, an ultrafilter on X is a set U consisting of subsets of X such that

1. The empty set is not an element of U

2. If A and B are subsets of X, A is a subset of B, and A is an element of U, then B is also an element of U.

3. If A and B are elements of U, then so is the intersection of A and B.

4. If A is a subset of X, then either A or X \ A is an element of U. (Note: axioms 1 and 3 imply that A and X \ Acannot both be elements of U.)

A characterization is given by the following theorem. A filter U on a set X is an ultrafilter if any of the followingconditions are true:

1. There is no filter F finer than U, i.e., U ⊆ F implies U = F.

2. A ∪B ∈ U implies A ∈ U or B ∈ U .

3. ∀A ⊆ X : A ∈ U or X \A ∈ U .

Another way of looking at ultrafilters on a set X is to define a function m on the power set of X by setting m(A) =1 if A is an element of U and m(A) = 0 otherwise. Such a function is called a 2-valued morphism. Then m is a

245

246 CHAPTER 69. ULTRAFILTER

finitely additive measure on X, and every property of elements of X is either true almost everywhere or false almosteverywhere. Note that this does not define a measure in the usual sense, which is required to be countably additive.For a filter F that is not an ultrafilter, one would say m(A) = 1 if A ∈ F and m(A) = 0 if X \ A ∈ F, leaving m undefinedelsewhere.A simple example of an ultrafilter is a principal ultrafilter, which consists of subsets of X that contain a given elementx of X. All ultrafilters on a finite set are principal.

69.2 Completeness

The completeness of an ultrafilter U on a set is the smallest cardinal κ such that there are κ elements of U whoseintersection is not in U. The definition implies that the completeness of any ultrafilter is at least ℵ0 . An ultrafilterwhose completeness is greater than ℵ0 —that is, the intersection of any countable collection of elements of U is stillin U—is called countably complete or σ -complete.The completeness of a countably complete nonprincipal ultrafilter on a set is always a measurable cardinal.

69.3 Generalization to partial orders

In order theory, an ultrafilter is a subset of a partially ordered set (a poset) that is maximal among all proper filters.Formally, this states that any filter that properly contains an ultrafilter has to be equal to the whole poset.

69.4 Special case: Boolean algebra

An important special case of the concept occurs if the considered poset is a Boolean algebra, as in the case of anultrafilter on a set (defined as a filter of the corresponding powerset). In this case, ultrafilters are characterized bycontaining, for each element a of the Boolean algebra, exactly one of the elements a and ¬a (the latter being theBoolean complement of a).Ultrafilters on a Boolean algebra can be identified with prime ideals, maximal ideals, and homomorphisms to the2-element Boolean algebra true, false, as follows:

• Maximal ideals of a Boolean algebra are the same as prime ideals.

• Given a homomorphism of a Boolean algebra onto true, false, the inverse image of “true” is an ultrafilter,and the inverse image of “false” is a maximal ideal.

• Given a maximal ideal of a Boolean algebra, its complement is an ultrafilter, and there is a unique homomor-phism onto true, false taking the maximal ideal to “false”.

• Given an ultrafilter of a Boolean algebra, its complement is a maximal ideal, and there is a unique homomor-phism onto true, false taking the ultrafilter to “true”.

Let us see another theorem, which could be used for the definition of the concept of “ultrafilter”. Let B denote aBoolean algebra and F a proper filter[1] in it. F is an ultrafilter iff:

for all a, b ∈ B , if a ∨ b ∈ F , then a ∈ F or b ∈ F

(To avoid confusion: the sign∨ denotes the join operation of the Boolean algebra, and logical connectives are renderedby English circumlocutions.) See details (and proof) in.[2]

69.5 Types and existence of ultrafilters

There are two very different types of ultrafilter: principal and free. A principal (or fixed, or trivial) ultrafilter is afilter containing a least element. Consequently, principal ultrafilters are of the form Fa = x | a ≤ x for some (but

69.6. APPLICATIONS 247

not all) elements a of the given poset. In this case a is called the principal element of the ultrafilter. For the case ofultrafilters on sets, the elements that qualify as principals are exactly the one-element sets. Thus, a principal ultrafilteron a set S consists of all sets containing a particular point of S. An ultrafilter on a finite set is principal. Any ultrafilterthat is not principal is called a free (or non-principal) ultrafilter.Note that an ultrafilter on an infinite set S is non-principal if and only if it contains the Fréchet filter of cofinite subsetsof S. This is obvious, since a non-principal ultrafilter contains no finite set, it means that, by taking complements, itcontains all cofinite subsets of S, which is exactly the Fréchet filter.One can show that every filter of a Boolean algebra (or more generally, any subset with the finite intersection property)is contained in an ultrafilter (see Ultrafilter lemma) and that free ultrafilters therefore exist, but the proofs involve theaxiom of choice (AC) in the form of Zorn’s Lemma. On the other hand, the statement that every filter is containedin an ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-knownintermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by theaxiom of choice (ZFC). Proofs involving the axiom of choice do not produce explicit examples of free ultrafilters.Nonetheless, almost all ultrafilters on an infinite set are free. By contrast, every ultrafilter of a finite poset (or on afinite set) is principal, since any finite filter has a least element. In ZF without the axiom of choice, it is possible thatevery ultrafilter is principal.

69.6 Applications

Ultrafilters on sets are useful in topology, especially in relation to compact Hausdorff spaces, and in model theoryin the construction of ultraproducts and ultrapowers. Every ultrafilter on a compact Hausdorff space converges toexactly one point. Likewise, ultrafilters on posets are most important if the poset is a Boolean algebra, since in thiscase the ultrafilters coincide with the prime filters. Ultrafilters in this form play a central role in Stone’s representationtheorem for Boolean algebras.The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely related to theabove-mentioned representation theorem. For any element a of P, let Da = U ∈ G | a ∈ U. This is most usefulwhen P is again a Boolean algebra, since in this situation the set of all Da is a base for a compact Hausdorff topologyon G. Especially, when considering the ultrafilters on a set S (i.e., the case that P is the powerset of S ordered viasubset inclusion), the resulting topological space is the Stone–Čech compactification of a discrete space of cardinality|S|.The ultraproduct construction in model theory uses ultrafilters to produce elementary extensions of structures. Forexample, in constructing hyperreal numbers as an ultraproduct of the real numbers, we first extend the domain ofdiscourse from the real numbers to sequences of real numbers. This sequence space is regarded as a superset ofthe reals by identifying each real with the corresponding constant sequence. To extend the familiar functions andrelations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this wouldlose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead we definethe functions and relations “pointwise modulo U", where U is an ultrafilter on the index set of the sequences; by Łoś'theorem, this preserves all properties of the reals that can be stated in first-order logic. If U is nonprincipal, then theextension thereby obtained is nontrivial.In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group. This con-struction yields a rigorous way to consider looking at the group from infinity, that is the large scale geometry of thegroup. Asymptotic cones are particular examples of ultralimits of metric spaces.Gödel’s ontological proof of God’s existence uses as an axiom that the set of all “positive properties” is an ultrafilter.In social choice theory, non-principal ultrafilters are used to define a rule (called a social welfare function) for ag-gregating the preferences of infinitely many individuals. Contrary to Arrow’s impossibility theorem for finitely manyindividuals, such a rule satisfies the conditions (properties) that Arrow proposes (e.g., Kirman and Sondermann,1972[3]). Mihara (1997,[4] 1999[5]) shows, however, such rules are practically of limited interest to social scientists,since they are non-algorithmic or non-computable.

248 CHAPTER 69. ULTRAFILTER

69.7 Ordering on ultrafilters

Rudin–Keisler ordering is a preorder on the class of ultrafilters defined as follows: if U is an ultrafilter on X, and Van ultrafilter on Y, then V ≤RK U if and only if there exists a function f: X → Y such that

C ∈ V ⇐⇒ f−1[C] ∈ U

for every subset C of Y.Ultrafilters U and V are Rudin–Keisler equivalent, U ≡RK V , if there exist sets A ∈ U , B ∈ V , and a bijectionf: A → B that satisfies the condition above. (If X and Y have the same cardinality, the definition can be simplified byfixing A = X, B = Y.)It is known that ≡RK is the kernel of ≤RK , i.e., U ≡RK V if and only if U ≤RK V and V ≤RK U .

69.8 Ultrafilters on ω

There are several special properties that an ultrafilter on ω may possess, which prove useful in various areas of settheory and topology.

• A non-principal ultrafilter U is a P-point (or weakly selective) iff for every partition of ω, Cn | n < ωsuch that Cn ∈ U,∀n < ω , there exists A ∈ U such that |A ∩ Cn| < ω, ∀n < ω .

• A non-principal ultrafilter U is Ramsey (or selective) iff for every partition of ω, Cn | n < ω such thatCn ∈ U,∀n < ω , there exists A ∈ U such that |A ∩ Cn| = 1, ∀n < ω

It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesisimplies the existence of Ramsey ultrafilters.[6] In fact, many hypotheses imply the existence of Ramsey ultrafilters,including Martin’s axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters.[7]

Therefore the existence of these types of ultrafilters is independent of ZFC.P-points are called as such because they are topological P-points in the usual topology of the space βω \ ω of non-principal ultrafilters. The name Ramsey comes from Ramsey’s theorem. To see why, one can prove that an ultrafilteris Ramsey if and only if for every 2-coloring of [ω]2 there exists an element of the ultrafilter that has a homogeneouscolor.An ultrafilter on ω is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal ultrafilters.

69.9 See also• Universal net

69.10 Notes[1] I.e., a filter F with the surplus restriction 0 /∈ F , i.e., being a filter that does not “degenerate” to coincide with the whole

(universe of the) Boolean algebra

[2] A Course in Universal Algebra (written by Stanley N. Burris and H.P. Sankappanavar), Corollary 3.13 on p. 149.

[3] Kirman, A.; Sondermann, D. (1972). “Arrow’s theorem, many agents, and invisible dictators”. Journal of Economic Theory5: 267. doi:10.1016/0022-0531(72)90106-8.

[4] Mihara, H. R. (1997). “Arrow’s Theorem and Turing computability” (PDF).Economic Theory 10 (2): 257–276. doi:10.1007/s001990050157Reprintedin K. V. Velupillai , S. Zambelli, and S. Kinsella, ed., Computable Economics, International Library of Critical Writingsin Economics, Edward Elgar, 2011.

[5] Mihara, H. R. (1999). “Arrow’s theorem, countably many agents, and more visible invisible dictators”. Journal of Mathe-matical Economics 32: 267–277. doi:10.1016/S0304-4068(98)00061-5.

69.11. REFERENCES 249

[6] Rudin, Walter (1956), “Homogeneity problems in the theory of Čech compactifications”, Duke Mathematical Journal 23(3): 409–419, doi:10.1215/S0012-7094-56-02337-7

[7] Wimmers, Edward (March 1982), “The Shelah P-point independence theorem”, Israel Journal of Mathematics (HebrewUniversity Magnes Press) 43 (1): 28–48, doi:10.1007/BF02761683

69.11 References• Comfort, W. W. (1977), “Ultrafilters: some old and some new results”, Bulletin of the American MathematicalSociety 83 (4): 417–455, doi:10.1090/S0002-9904-1977-14316-4, ISSN 0002-9904, MR 0454893

• Comfort, W. W.; Negrepontis, S. (1974), The theory of ultrafilters, Berlin, New York: Springer-Verlag, MR0396267

Chapter 70

Union-closed sets conjecture

In combinatorial mathematics, the union-closed sets conjecture is an elementary problem, posed by Péter Frankl in1979 and still open. A family of sets is said to be union-closed if the union of any two sets from the family remains inthe family. The conjecture states that for any finite union-closed family of finite sets, other than the family consistingonly of the empty set, there exists an element that belongs to at least half of the sets in the family.

70.1 Equivalent forms

If F is a union-closed family of sets, the family of complement sets to sets in F is closed under intersection, andan element that belongs to at least half of the sets of F belongs to at most half of the complement sets. Thus, anequivalent form of the conjecture (the form in which it was originally stated) is that, for any intersection-closed familyof sets that contains more than one set, there exists an element that belongs to at most half of the sets in the family.Although stated above in terms of families of sets, Frankl’s conjecture has also been formulated and studied as aquestion in lattice theory. A lattice is a partially ordered set in which for two elements x and y there is a uniquegreatest element less than or equal to both of them (the meet of x and y) and a unique least element greater than orequal to both of them (the join of x and y). The family of all subsets of a set S, ordered by set inclusion, forms a latticein which the meet is represented by the set-theoretic intersection and the join is represented by the set-theoretic union;a lattice formed in this way is called a Boolean lattice. The lattice-theoretic version of Frankl’s conjecture is that inany finite lattice there exists an element x that is not the join of any two smaller elements, and such that the number ofelements greater than or equal to x totals at most half the lattice, with equality only if the lattice is a Boolean lattice.As Abe (2000) shows, this statement about lattices is equivalent to the Frankl conjecture for union-closed sets: eachlattice can be translated into a union-closed set family, and each union-closed set family can be translated into alattice, such that the truth of the Frankl conjecture for the translated object implies the truth of the conjecture for theoriginal object. This lattice-theoretic version of the conjecture is known to be true for several natural subclasses oflattices (Abe 2000; Poonen 1992; Reinhold 2000) but remains open in the general case.

70.2 Families known to satisfy the conjecture

The conjecture has been proven for many special cases of union-closed set families. In particular, it is known to betrue for

• families of at most 46 sets (Roberts & Simpson 2010).

• families of sets such that their union has at most 11 elements,[1] and

• families of sets in which the smallest set has one or two elements.[2]

250

70.3. HISTORY 251

70.3 History

Péter Frankl stated the conjecture, in terms of intersection-closed set families, in 1979, and so the conjecture isusually credited to him and sometimes called the Frankl conjecture. The earliest publication of the union-closedversion of the conjecture appears to be by Duffus (1985).

70.4 Notes[1] Bošnjak and Marković (2008), improving previous bounds by Morris (2006), Lo Faro (1994) and others.

[2] Sarvate and Renaud (1989), since rediscovered by several other authors. If a one-element or two-element set S exists, someelement of S belongs to at least half the sets in the family, but the same property does not hold for three-element sets, dueto counterexamples of Sarvate, Renaud, and Ronald Graham.

70.5 References• Abe, Tetsuya (2000). “Strong semimodular lattices and Frankl’s conjecture”. Algebra Universalis 44 (3–4):

379–382. doi:10.1007/s000120050195.

• Bošnjak, Ivica; Marković, Peter (2008). “The 11-element case of Frankl’s conjecture”. Electronic Journal ofCombinatorics 15 (1): R88.

• Duffus, D. (1985). Rival, I., ed. Open problem session. Graphs and Order. D. Reidel. p. 525.

• Lo Faro, Giovanni (1994). “Union-closed sets conjecture: improved bounds”. J. Combin. Math. Combin.Comput. 16: 97–102. MR 1301213.

• Morris, Robert (2006). “FC-families and improved bounds for Frankl’s conjecture”. European Journal ofCombinatorics 27 (2): 269–282. doi:10.1016/j.ejc.2004.07.012. MR 2199779.

• Poonen, Bjorn (1992). “Union-closed families”. Journal of Combinatorial Theory, Series A 59 (2): 253–268.doi:10.1016/0097-3165(92)90068-6. MR 1149898.

• Reinhold, Jürgen (2000). “Frankl’s conjecture is true for lower semimodular lattices”. Graphs Combin. 16 (1):115–116. doi:10.1007/s003730050008.

• Roberts, Ian; Simpson, Jamie (2010). “A note on the union-closed sets conjecture” (PDF). Australas. J.Combin. 47: 265–267.

• Sarvate, D. G.; Renaud, J.-C. (1989). “On the union-closed sets conjecture”. Ars Combin. 27: 149–153. MR0989460.

70.6 External links• Frankl’s union-closed sets conjecture, the Open Problem Garden.

• Union-Closed Sets Conjecture (1979). In Open Problems - Graph Theory and Combinatorics, collected by D.B. West.

• Weisstein, Eric W., “Union-Closed Sets Conjecture”, MathWorld.

Chapter 71

Universal set

For other uses, see Universal set (disambiguation).

In set theory, a universal set is a set which contains all objects, including itself.[1] In set theory as usually formulated,the conception of a universal set leads to a paradox (Russell’s paradox) and is consequently not allowed. However,some non-standard variants of set theory include a universal set. It is often symbolized by the Greek letter xi.

71.1 Reasons for nonexistence

Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, donot allow for the existence of a universal set. Its existence would cause paradoxes which would make the theoryinconsistent.

71.1.1 Russell’s paradox

Russell’s paradox prevents the existence of a universal set in Zermelo–Fraenkel set theory and other set theories thatinclude Zermelo's axiom of comprehension. This axiom states that, for any formula φ(x) and any set A, there existsanother set

x ∈ A | φ(x)

that contains exactly those elements x of A that satisfy φ . If a universal set V existed and the axiom of comprehensioncould be applied to it, then there would also exist another set x ∈ V | x ∈ x , the set of all sets that do not containthemselves. However, as Bertrand Russell observed, this set is paradoxical. If it contains itself, then it should notcontain itself, and vice versa. For this reason, it cannot exist.

71.1.2 Cantor’s theorem

A second difficulty with the idea of a universal set concerns the power set of the set of all sets. Because this power setis a set of sets, it would automatically be a subset of the set of all sets, provided that both exist. However, this conflictswith Cantor’s theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality thanthe set itself.

71.2 Theories of universality

The difficulties associated with a universal set can be avoided either by using a variant of set theory in which theaxiom of comprehension is restricted in some way, or by using a universal object that is not considered to be a set.

252

71.3. NOTES 253

71.2.1 Restricted comprehension

There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V doesexist (and V ∈ V is true). In these theories, Zermelo’s axiom of comprehension does not hold in general, and theaxiom of comprehension of naive set theory is restricted in a different way. A set theory containing a universal set isnecessarily a non-well-founded set theory.The most widely studied set theory with a universal set is Willard Van Orman Quine’s New Foundations. AlonzoChurch and Arnold Oberschelp also published work on such set theories. Church speculated that his theory mightbe extended in a manner consistent with Quine’s,[2] but this is not possible for Oberschelp’s, since in it the singletonfunction is provably a set,[3] which leads immediately to paradox in New Foundations.[4] The most recent advancesin this area have been made by Randall Holmes who published an online draft version of the book Elementary SetTheory with a Universal Set in 2012.[5]

71.2.2 Universal objects that are not sets

Main article: Universe (mathematics)

The idea of a universal set seems intuitively desirable in the Zermelo–Fraenkel set theory, particularly because mostversions of this theory do allow the use of quantifiers over all sets (see universal quantifier). One way of allowingan object that behaves similarly to a universal set, without creating paradoxes, is to describe V and similar largecollections as proper classes rather than as sets. One difference between a universal set and a universal class is thatthe universal class does not contain itself, because proper classes cannot be elements of other classes. Russell’sparadox does not apply in these theories because the axiom of comprehension operates on sets, not on classes.The category of sets can also be considered to be a universal object that is, again, not itself a set. It has all sets aselements, and also includes arrows for all functions from one set to another. Again, it does not contain itself, becauseit is not itself a set.

71.3 Notes[1] Forster 1995 p. 1.

[2] Church 1974 p. 308. See also Forster 1995 p. 136 or 2001 p. 17.

[3] Oberschelp 1973 p. 40.

[4] Holmes 1998 p. 110.

[5] http://math.boisestate.edu/~holmes/

71.4 References• Alonzo Church (1974). “Set Theory with a Universal Set,” Proceedings of the Tarski Symposium. Proceedingsof Symposia in Pure Mathematics XXV, ed. L. Henkin, American Mathematical Society, pp. 297–308.

• T. E. Forster (1995). Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford Logic Guides31). Oxford University Press. ISBN 0-19-851477-8.

• T. E. Forster (2001). “Church’s Set Theory with a Universal Set.”• Bibliography: Set Theory with a Universal Set, originated by T. E. Forster and maintained by Randall Holmes

at Boise State University.• Randall Holmes (1998). Elementary Set theory with a Universal Set, volume 10 of the Cahiers du Centre de

Logique, Academia, Louvain-la-Neuve (Belgium).• Arnold Oberschelp (1973). “Set Theory over Classes,” Dissertationes Mathematicae 106.• Willard Van Orman Quine (1937) “New Foundations for Mathematical Logic,”AmericanMathematicalMonthly

44, pp. 70–80.

254 CHAPTER 71. UNIVERSAL SET

71.5 External links• Weisstein, Eric W., “Universal Set”, MathWorld.

Chapter 72

Universe (mathematics)

In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains(as elements) all the entities one wishes to consider in a given situation. There are several versions of this generalidea, described in the following sections.

72.1 In a specific context

Perhaps the simplest version is that any set can be a universe, so long as the object of study is confined to that particularset. If the object of study is formed by the real numbers, then the real line R, which is the real number set, could bethe universe under consideration. Implicitly, this is the universe that Georg Cantor was using when he first developedmodern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis. The only sets thatCantor was originally interested in were subsets of R.This concept of a universe is reflected in the use of Venn diagrams. In a Venn diagram, the action traditionally takesplace inside a large rectangle that represents the universe U. One generally says that sets are represented by circles;but these sets can only be subsets of U. The complement of a set A is then given by that portion of the rectangleoutside of A's circle. Strictly speaking, this is the relative complement U \ A of A relative to U; but in a contextwhere U is the universe, it can be regarded as the absolute complement AC of A. Similarly, there is a notion of thenullary intersection, that is the intersection of zero sets (meaning no sets, not null sets). Without a universe, thenullary intersection would be the set of absolutely everything, which is generally regarded as impossible; but with theuniverse in mind, the nullary intersection can be treated as the set of everything under consideration, which is simplyU.These conventions are quite useful in the algebraic approach to basic set theory, based on Boolean lattices. Exceptin some non-standard forms of axiomatic set theory (such as New Foundations), the class of all sets is not a Booleanlattice (it is only a relatively complemented lattice). In contrast, the class of all subsets of U, called the power setof U, is a Boolean lattice. The absolute complement described above is the complement operation in the Booleanlattice; and U, as the nullary intersection, serves as the top element (or nullary meet) in the Boolean lattice. Then DeMorgan’s laws, which deal with complements of meets and joins (which are unions in set theory) apply, and applyeven to the nullary meet and the nullary join (which is the empty set).

72.2 In ordinary mathematics

However, once subsets of a given set X (in Cantor’s case, X = R) are considered, the universe may need to be a setof subsets of X. (For example, a topology on X is a set of subsets of X.) The various sets of subsets of X will notthemselves be subsets of X but will instead be subsets of PX, the power set of X. This may be continued; the objectof study may next consist of such sets of subsets of X, and so on, in which case the universe will be P(PX). In anotherdirection, the binary relations on X (subsets of the Cartesian product X × X) may be considered, or functions from Xto itself, requiring universes like P(X × X) or XX.Thus, even if the primary interest is X, the universe may need to be considerably larger than X. Following the aboveideas, one may want the superstructure overX as the universe. This can be defined by structural recursion as follows:

255

256 CHAPTER 72. UNIVERSE (MATHEMATICS)

• Let S0X be X itself.

• Let S1X be the union of X and PX.

• Let S2X be the union of S1X and P(S1X).

• In general, let Sn₊₁X be the union of S X and P(SnX).

Then the superstructure over X, written SX, is the union of S0X, S1X, S2X, and so on; or

SX :=

∞∪i=0

SiX .

Note that no matter what set X is the starting point, the empty set will belong to S1X. The empty set is the vonNeumann ordinal [0]. Then [0], the set whose only element is the empty set, will belong to S2X; this is the vonNeumann ordinal [1]. Similarly, [1] will belong to S3X, and thus so will [0],[1], as the union of [0] and[1]; this is the von Neumann ordinal [2]. Continuing this process, every natural number is represented in thesuperstructure by its von Neumann ordinal. Next, if x and y belong to the superstructure, then so does x,x,y,which represents the ordered pair (x,y). Thus the superstructure will contain the various desired Cartesian products.Then the superstructure also contains functions and relations, since these may be represented as subsets of Cartesianproducts. The process also gives ordered n-tuples, represented as functions whose domain is the von Neumann ordinal[n]. And so on.So if the starting point is just X = , a great deal of the sets needed for mathematics appear as elements of thesuperstructure over . But each of the elements of S will be finite sets! Each of the natural numbers belongsto it, but the set N of all natural numbers does not (although it is a subset of S). In fact, the superstructure over consists of all of the hereditarily finite sets. As such, it can be considered the universe of finitist mathematics.Speaking anachronistically, one could suggest that the 19th-century finitist Leopold Kronecker was working in thisuniverse; he believed that each natural number existed but that the set N (a "completed infinity") did not.However, S is unsatisfactory for ordinary mathematicians (who are not finitists), because even though N may beavailable as a subset of S, still the power set ofN is not. In particular, arbitrary sets of real numbers are not available.So it may be necessary to start the process all over again and form S(S). However, to keep things simple, one cantake the set N of natural numbers as given and form SN, the superstructure over N. This is often considered theuniverse of ordinary mathematics. The idea is that all of the mathematics that is ordinarily studied refers to elementsof this universe. For example, any of the usual constructions of the real numbers (say by Dedekind cuts) belongs toSN. Even non-standard analysis can be done in the superstructure over a non-standard model of the natural numbers.One should note a slight shift in philosophy from the previous section, where the universe was any set U of interest.There, the sets being studied were subsets of the universe; now, they are members of the universe. Thus althoughP(SX) is a Boolean lattice, what is relevant is that SX itself is not. Consequently, it is rare to apply the notions ofBoolean lattices and Venn diagrams directly to the superstructure universe as they were to the power-set universesof the previous section. Instead, one can work with the individual Boolean lattices PA, where A is any relevant setbelonging to SX; then PA is a subset of SX (and in fact belongs to SX). In Cantor’s case X = R in particular, arbitrarysets of real numbers are not available, so there it may indeed be necessary to start the process all over again.

72.3 In set theory

It is possible to give a precise meaning to the claim that SN is the universe of ordinary mathematics; it is a model ofZermelo set theory, the axiomatic set theory originally developed by Ernst Zermelo in 1908. Zermelo set theory wassuccessful precisely because it was capable of axiomatising “ordinary” mathematics, fulfilling the programme begunby Cantor over 30 years earlier. But Zermelo set theory proved insufficient for the further development of axiomaticset theory and other work in the foundations of mathematics, especially model theory. For a dramatic example, thedescription of the superstructure process above cannot itself be carried out in Zermelo set theory! The final step,forming S as an infinitary union, requires the axiom of replacement, which was added to Zermelo set theory in 1922to form Zermelo–Fraenkel set theory, the set of axioms most widely accepted today. So while ordinary mathematicsmay be done in SN, discussion of SN goes beyond the “ordinary”, into metamathematics.

72.4. IN CATEGORY THEORY 257

But if high-powered set theory is brought in, the superstructure process above reveals itself to be merely the beginningof a transfinite recursion. Going back to X = , the empty set, and introducing the (standard) notation Vi for Si,V0 = , V1 = P, and so on as before. But what used to be called “superstructure” is now just the next item on thelist: Vω, where ω is the first infinite ordinal number. This can be extended to arbitrary ordinal numbers:

Vi :=∪j<i

PVj

defines Vi for any ordinal number i. The union of all of the Vi is the von Neumann universe V:

V :=∪i

Vi

Note that every individual Vi is a set, but their union V is a proper class. The axiom of foundation, which was addedto ZF set theory at around the same time as the axiom of replacement, says that every set belongs to V.

Kurt Gödel's constructible universe L and the axiom of constructibility

Inaccessible cardinals yield models of ZF and sometimes additional axioms, and are equivalent to theexistence of the Grothendieck universe set

72.4 In category theory

There is another approach to universes which is historically connected with category theory. This is the idea of aGrothendieck universe. Roughly speaking, a Grothendieck universe is a set inside which all the usual operations ofset theory can be performed. This version of a universe is defined to be any set for which the following axioms hold:[1]

1. x ∈ u ∈ U implies x ∈ U

2. u ∈ U and v ∈ U imply u,v, (u,v), and u× v ∈ U .

3. x ∈ U implies Px ∈ U and ∪x ∈ U

4. ω ∈ U (here ω = 0, 1, 2, ... is the set of all finite ordinals.)

5. if f : a→ b is a surjective function with a ∈ U and b ⊂ U , then b ∈ U .

The advantage of a Grothendieck universe is that it is actually a set, and never a proper class. The disadvantage isthat if one tries hard enough, one can leave a Grothendieck universe.The most common use of a Grothendieck universe U is to take U as a replacement for the category of all sets. Onesays that a set S is U-small if S ∈U, and U-large otherwise. The category U-Set of all U-small sets has as objects allU-small sets and as morphisms all functions between these sets. Both the object set and the morphism set are sets,so it becomes possible to discuss the category of “all” sets without invoking proper classes. Then it becomes possibleto define other categories in terms of this new category. For example, the category of all U-small categories is thecategory of all categories whose object set and whose morphism set are in U. Then the usual arguments of set theoryare applicable to the category of all categories, and one does not have to worry about accidentally talking about properclasses. Because Grothendieck universes are extremely large, this suffices in almost all applications.Often when working with Grothendieck universes, mathematicians assume the Axiom of Universes: “For any set x,there exists a universe U such that x ∈U.” The point of this axiom is that any set one encounters is then U-small forsome U, so any argument done in a general Grothendieck universe can be applied. This axiom is closely related tothe existence of strongly inaccessible cardinals.

Set-like toposes

258 CHAPTER 72. UNIVERSE (MATHEMATICS)

72.5 See also• Herbrand universe

• Free object

72.6 Notes[1] Mac Lane 1998, p.22

72.7 References• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag New York, Inc.

72.8 External links• Hazewinkel, Michiel, ed. (2001), “Universe”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

• Weisstein, Eric W., “Universal Set”, MathWorld.

Chapter 73

Vietoris–Rips complex

A Vietoris–Rips complex of a set of 23 points in the Euclidean plane. This complex has sets of up to four points: the points themselves(shown as red circles), pairs of points (black edges), triples of points (pale blue triangles), and quadruples of points (dark bluetetrahedrons).

In topology, theVietoris–Rips complex, also called theVietoris complex orRips complex, is an abstract simplicialcomplex that can be defined from any metric spaceM and distance δ by forming a simplex for every finite set of pointsthat has diameter at most δ. That is, it is a family of finite subsets of M, in which we think of a subset of k pointsas forming a (k − 1)-dimensional simplex (an edge for two points, a triangle for three points, a tetrahedron for fourpoints, etc.); if a finite set S has the property that the distance between every pair of points in S is at most δ, then weinclude S as a simplex in the complex.

259

260 CHAPTER 73. VIETORIS–RIPS COMPLEX

73.1 History

The Vietoris–Rips complex was originally called the Vietoris complex, for Leopold Vietoris, who introduced it as ameans of extending homology theory from simplicial complexes to metric spaces.[1] After Eliyahu Rips applied thesame complex to the study of hyperbolic groups, its use was popularized by Gromov (1987), who called it the Ripscomplex.[2] The name “Vietoris–Rips complex” is due to Hausmann (1995).[3]

73.2 Relation to Čech complex

The Vietoris–Rips complex is closely related to the Čech complex (or nerve) of a set of balls, which has a simplex forevery finite subset of balls with nonempty intersection: in a geodesically convex space Y, the Vietoris–Rips complexof any subspace X ⊂ Y for distance δ has the same points and edges as the Čech complex of the set of balls of radiusδ/2 in Y that are centered at the points of X. However, unlike the Čech complex, the Vietoris–Rips complex of Xdepends only on the intrinsic geometry of X, and not on any embedding of X into some larger space.As an example, consider the uniform metric space M3 consisting of three points, each at unit distance from eachother. The Vietoris–Rips complex of M3, for δ = 1, includes a simplex for every subset of points in M3, including atriangle for M3 itself. If we embed M3 as an equilateral triangle in the Euclidean plane, then the Čech complex ofthe radius-1/2 balls centered at the points of M3 would contain all other simplexes of the Vietoris–Rips complex butwould not contain this triangle, as there is no point of the plane contained in all three balls. However, if M3 is insteadembedded into a metric space that contains a fourth point at distance 1/2 from each of the three points of M3, theČech complex of the radius-1/2 balls in this space would contain the triangle. Thus, the Čech complex of fixed-radiusballs centered at M3 differs depending on which larger space M3 might be embedded into, while the Vietoris–Ripscomplex remains unchanged.If any metric space X is embedded in an injective metric space Y, the Vietoris–Rips complex for distance δ and Xcoincides with the Čech complex of the balls of radius δ/2 centered at the points of X in Y. Thus, the Vietoris–Ripscomplex of any metric space M equals the Čech complex of a system of balls in the tight span of M.

73.3 Relation to unit disk graphs and clique complexes

The Vietoris–Rips complex for δ = 1 contains an edge for every pair of points that are at unit distance or less in thegiven metric space. As such, its 1-skeleton is the unit disk graph of its points. It contains a simplex for every cliquein the unit disk graph, so it is the clique complex or flag complex of the unit disk graph.[4] More generally, the cliquecomplex of any graphG is a Vietoris–Rips complex for the metric space having as points the vertices ofG and havingas its distances the lengths of the shortest paths in G.

73.4 Other results

If M is a closed Riemannian manifold, then for sufficiently small values of δ the Vietoris–Rips complex of M, or ofspaces sufficiently close to M, is homotopy equivalent to M itself.[5]

Chambers, Erickson & Worah (2008) describe efficient algorithms for determining whether a given cycle is con-tractible in the Rips complex of any finite point set in the Euclidean plane.

73.5 Applications

As with unit disk graphs, the Vietoris–Rips complex has been applied in computer science to model the topologyof ad hoc wireless communication networks. One advantage of the Vietoris–Rips complex in this application is thatit can be determined only from the distances between the communication nodes, without having to infer their exactphysical locations. A disadvantage is that, unlike the Čech complex, the Vietoris–Rips complex does not directlyprovide information about gaps in communication coverage, but this flaw can be ameliorated by sandwiching theČech complex between two Vietoris–Rips complexes for different values of δ.[6]

73.6. NOTES 261

Vietoris–Rips complexes have also been applied for feature-extraction in digital image data; in this application, thecomplex is built from a high-dimensional metric space in which the points represent low-level image features.[7]

73.6 Notes[1] Vietoris (1927); Lefschetz (1942); Hausmann (1995); Reitberger (2002).

[2] Hausmann (1995); Reitberger (2002).

[3] Reitberger (2002).

[4] Chambers, Erickson & Worah (2008).

[5] Hausmann (1995), Latschev (2001).

[6] de Silva & Ghrist (2006), Muhammad & Jadbabaie (2007).

[7] Carlsson, Carlsson & de Silva (2006).

73.7 References• Carlsson, E.; Carlsson, G.; de Silva, V. (2006), “An algebraic topological method for feature identification”,Int. J. Comput. Geom. Appl. 16 (4): 291–314, doi:10.1142/S021819590600204X.

• Chambers, Erin W.; Erickson, Jeff; Worah, Pratik (2008), “Testing contractibility in planar Rips complexes”,Proceedings of the 24th Annual ACMSymposium onComputational Geometry, pp. 251–259, doi:10.1145/1377676.1377721.

• Chazal, Frédéric; Oudot, Steve (2008), “Towards Persistence-Based Reconstruction in Euclidean Spaces”,ACM Symposium on Computational Geometry: 232–241, arXiv:0712.2638, doi:10.1145/1377676.1377719,ISBN 978-1-60558-071-5.

• de Silva, V.; Ghrist, R. (2006), “Coordinate-free coverage in sensor networks with controlled boundaries via ho-mology”, The International Journal of Robotics Research 25 (12): 1205–1222, doi:10.1177/0278364906072252.

• Gromov, M. (1987), “Hyperbolic groups”, Essays in group theory, Mathematical Sciences Research InstitutePublications 8, Springer-Verlag, pp. 75–263.

• Hausmann, J.-C. (1995), “On the Vietoris–Rips complexes and a cohomology theory for metric spaces”,Prospects in Topology: Proceedings of a conference in honour of William Browder, Annals of MathematicsStudies 138, Princeton Univ. Press, pp. 175–188, MR 1368659.

• Latschev, Janko (2001), “Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold”,Archiv der Mathematik 77 (6): 522–528, doi:10.1007/PL00000526, MR 1879057.

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Chapter 74

∞-groupoid

In category theory, a branch of mathematics, an ∞-groupoid (also called Kan complex) is a fibrant object in thecategory of simplicial sets (with the standard model structure). It is an ∞-category generalization of a groupoid, acategory in which every morphism is an isomorphism.The homotopy hypothesis states that ∞-groupoids are spaces.

74.1 See also• Groupoid

• Homotopy type theory

• Pursuing Stacks

74.2 External links• infinity-groupoid in nLab

262

74.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 263

74.3 Text and image sources, contributors, and licenses

74.3.1 Text• Abstract simplicial complex Source: https://en.wikipedia.org/wiki/Abstract_simplicial_complex?oldid=672208012 Contributors: Ax-

elBoldt, Zundark, Tomo, Charles Matthews, Altenmann, Giftlite, BenFrantzDale, Jkseppan, Zhen Lin, Zaslav, Gauge, Billlion, Fiedorow,Danhash, Oleg Alexandrov, Linas, BD2412, Gaius Cornelius, Hv, SmackBot, Henning Makholm, MichaelNChristoff, 345Kai, DavidEppstein, Trevorgoodchild, JackSchmidt, Saddhiyama, Rswarbrick, Yobot, Citation bot, LilHelpa, Queen Rhana, Stereospan, Tgoodwil,Gud music only, Helpful Pixie Bot, Lesser Cartographies, ProboscideaRubber15 and Anonymous: 10

• Almost disjoint sets Source: https://en.wikipedia.org/wiki/Almost_disjoint_sets?oldid=607152448 Contributors: Revolver, CharlesMatthews, Kaol, Nickj, Oleg Alexandrov, Pol098, Salix alba, SmackBot, Maksim-e~enwiki, Dreadstar, CBM, Addbot, Yobot, SassoBot,Erik9bot, Kasterma and Anonymous: 3

• Antimatroid Source: https://en.wikipedia.org/wiki/Antimatroid?oldid=678393170 Contributors: Altenmann, Giftlite, Peter Kwok, Za-slav, Algebraist, Wavelength, Wimt, Mhym, E-Kartoffel, Myasuda, Headbomb, D Haggerty, Pftupper, David Eppstein, R'n'B, Lantonov,LokiClock, Justin W Smith, DOI bot, Citation bot, Citation bot 1, Kiefer.Wolfowitz, CrowzRSA, Helpful Pixie Bot, Mark viking andAnonymous: 3

• Block design Source: https://en.wikipedia.org/wiki/Block_design?oldid=674635383 Contributors: Michael Hardy, Dominus, Docu, Sil-verfish, Charles Matthews, Giftlite, Rich Farmbrough, Xezbeth, Zaslav, Mairi, Kaganer, Joriki, Linas, Will Orrick, Btyner, Rjwilmsi,RFBailey, Ott2, Cullinane, Vicarious, SmackBot, Ttzz, Gutworth, Nbarth, Cícero, Rogério Brito, Cydebot, Flowerpotman, Headbomb,JustAGal, A3nm, David Eppstein, Maproom, Squids and Chips, Kmhkmh, VVVBot, Melcombe, Qwfp, Addbot, KorinoChikara, Mj-collins68, Nassrat, Yobot, AnomieBOT, Mendelso, Howard McCay, FrescoBot, Nageh, SlumdogAramis, Kiefer.Wolfowitz, RjwilmsiBot,RA0808, ClueBot NG, Wcherowi, Myas012, Helpful Pixie Bot, BattyBot, Illia Connell, Mnoshad and Anonymous: 15

• Carathéodory’s theorem (convex hull) Source: https://en.wikipedia.org/wiki/Carath%C3%A9odory’s_theorem_(convex_hull)?oldid=628980653 Contributors: The Anome, Michael Hardy, Charles Matthews, Dysprosia, Fibonacci, Tosha, Giftlite, Elroch, WpZurp, OlegAlexandrov, Linas, Ruud Koot, Hbf~enwiki, BD2412, Rjwilmsi, Brighterorange, YurikBot, Ott2, RDBury, BeteNoir, Reedy, Lambiam,A. Pichler, David Eppstein, Val001, Aff-conv, Addbot, Luckas-bot, Yobot, Citation bot, Ash4Math, Ivan Kuckir, Citation bot 1, Tkuvho,Kiefer.Wolfowitz, Xnn, Joerg Bader, Brad7777, Filedelinkerbot and Anonymous: 10

• Clique complex Source: https://en.wikipedia.org/wiki/Clique_complex?oldid=652127271 Contributors: Michael Hardy, YUL89YYZ,Igorpak, Marozols, Ott2, Chris the speller, Headbomb, David Eppstein, LarRan, Balabiot, Yobot, Citation bot 1, Helpful Pixie Bot, Ikaalinand Anonymous: 1

• Combinatorial design Source: https://en.wikipedia.org/wiki/Combinatorial_design?oldid=679675291 Contributors: The Anome, Ed-ward, Michael Hardy, Charles Matthews, Giftlite, Bender235, Igorpak, Will Orrick, Btyner, Rjwilmsi, Michael Slone, SmackBot, Ttzz,Nbarth, Cícero, Pladdin~enwiki, CBM, Cydebot, Hardmath, Lantonov, Melcombe, Niceguyedc, Nassrat, AnomieBOT, Carturo222,Kiefer.Wolfowitz, Xnn, John of Reading, Wcherowi, Helpful Pixie Bot, Melcous and Anonymous: 9

• Configuration (geometry) Source: https://en.wikipedia.org/wiki/Configuration_(geometry)?oldid=682413404Contributors: Tomo, MichaelHardy, Charles Matthews, Giftlite, Rich Farmbrough, Igorpak, R.e.b., Michael Kinyon, Headbomb, Wayiran, Steelpillow, David Eppstein,FANSTARbot, BotMultichill, Addbot, Blah28948, Citation bot, FrescoBot, Double sharp, RjwilmsiBot, Wcherowi, Helpful Pixie Bot,Solomon7968, Boodlepounce, Nathann.cohen and Anonymous: 4

• Content (measure theory) Source: https://en.wikipedia.org/wiki/Content_(measure_theory)?oldid=665512444 Contributors: MichaelHardy, Nandhp, Oleg Alexandrov, R.e.b., SmackBot, Decltype, Pascal.Tesson, Alaibot, Vanish2, Twistedliquidcrystal, BartekChom,ClueBot, Addbot, Yobot, Omnipaedista, Content (measure theory), Trappist the monk, John of Reading, ClueBot NG, Qetuth, Hierar-chivist and Anonymous: 3

• Dedekind number Source: https://en.wikipedia.org/wiki/Dedekind_number?oldid=676384491 Contributors: Michael Hardy, Roman-poet, Macrakis, Rjwilmsi, Chris the speller, CRGreathouse, FlyingToaster, HenningThielemann, Headbomb, David Eppstein, Gfis,Fharper1961, JimInTheUSA, Watchduck, Addbot, Luckas-bot, FrescoBot, Slawekb, Toshio Yamaguchi and Anonymous: 2

• Delta set Source: https://en.wikipedia.org/wiki/Delta_set?oldid=628564984 Contributors: Michael Hardy, Gabbe, Lethe, Rich Farm-brough, Rjwilmsi, Stephenb, Noian, CBM, David Eppstein, Mbw314, Jack-A-Roe, UnCatBot, Yobot, GB fan, FrescoBot, Citation bot 1,RjwilmsiBot, WikiCopter, Helpful Pixie Bot, ChrisGualtieri, Mark viking and Anonymous: 3

• Delta-ring Source: https://en.wikipedia.org/wiki/Delta-ring?oldid=544536467 Contributors: Charles Matthews, Touriste, Salix alba,Keith111, Addbot and ZéroBot

• Dendroidal set Source: https://en.wikipedia.org/wiki/Dendroidal_set?oldid=508150351 Contributors: Brianhe, Linas, Rjwilmsi, R.e.b.and Headbomb

• Discrete differential geometry Source: https://en.wikipedia.org/wiki/Discrete_differential_geometry?oldid=643128826 Contributors:Bearcat, CBM, David Eppstein, Rajpaj, GermanX, JohnBlackburne, Addbot, Tinus74, Delaszk, Yobot, Drilnoth, Charvest, Helpful PixieBot, Brad7777, Qetuth and Anonymous: 3

• Disjoint sets Source: https://en.wikipedia.org/wiki/Disjoint_sets?oldid=675919630 Contributors: AxelBoldt, Mav, Tarquin, Jeronimo,Shd~enwiki, Arvindn, Toby Bartels, Michael Hardy, Wshun, Revolver, Charles Matthews, Fibonacci, Robbot, Sheskar~enwiki, TobiasBergemann, Giftlite, Fropuff, Achituv~enwiki, Almit39, Nickj, Jumbuck, Dirac1933, Salix alba, FlaBot, Jameshfisher, Chobot, Alge-braist, YurikBot, RobotE, Bota47, Light current, Raijinili, SmackBot, RDBury, Maksim-e~enwiki, DHN-bot~enwiki, Vina-iwbot~enwiki,Typinaway, Mets501, Iridescent, Devourer09, Ezrakilty, Christian75, Kilva, RobHar, Salgueiro~enwiki, JAnDbot, David Eppstein,Miker70741, J.delanoy, STBotD, VolkovBot, Jamelan, PanagosTheOther, Alexbot, DumZiBoT, WikHead, Addbot, LaaknorBot, Sp-Bot, Legobot, Luckas-bot, Yobot, GrouchoBot, Erik9bot, MastiBot, Peacedance, Tbhotch, Ripchip Bot, ZéroBot, 28bot, ClueBot NG,Prakhar.agrwl, Helpful Pixie Bot, BG19bot, Amyxz, Solomon7968, JYBot, Stephan Kulla, Domez99, Flat Out and Anonymous: 34

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264 CHAPTER 74. ∞-GROUPOID

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• Family of sets Source: https://en.wikipedia.org/wiki/Family_of_sets?oldid=679067161 Contributors: Toby Bartels, Charles Matthews,Chris Howard, Oleg Alexandrov, Salix alba, Chobot, Wavelength, Arthur Rubin, Reedy, Mhss, CBM, RomanXNS, David Eppstein,JoergenB, Pomte, PixelBot, Avoided, Addbot, Matěj Grabovský, Calle, Erik9bot, DivineAlpha, NearSetAccount, Xnn, Sheerun, ClueBotNG, Wcherowi and Anonymous: 9

• Field of sets Source: https://en.wikipedia.org/wiki/Field_of_sets?oldid=674667676 Contributors: Charles Matthews, David Shay, To-bias Bergemann, Giftlite, William Elliot, Rich Farmbrough, Paul August, Touriste, DaveGorman, Kuratowski’s Ghost, Bart133, OlegAlexandrov, Salix alba, YurikBot, Trovatore, Mike Dillon, Arthur Rubin, That Guy, From That Show!, SmackBot, Mhss, Gala.martin,Stotr~enwiki, Mathematrucker, R'n'B, Lamro, BotMultichill, VVVBot, Hans Adler, Addbot, DaughterofSun, Jarble, AnomieBOT, Cita-tion bot, Kiefer.Wolfowitz, Yahia.barie, EmausBot, Tijfo098, ClueBot NG, MerlIwBot, BattyBot, Deltahedron, Mohammad Abubakarand Anonymous: 14

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• Generalized quadrangle Source: https://en.wikipedia.org/wiki/Generalized_quadrangle?oldid=666528896Contributors: Charles Matthews,Oleg Alexandrov, Mathbot, Evilbu, SmackBot, Michael Kinyon, STBot, Leyo, Rocchini, Kthas, Wikikane, Addbot, Laurinavicius, Yobot,JackieBot, Octonion, ClueBot NG, Wcherowi, BG19bot, Anurag Bishnoi and Anonymous: 9

• Greedoid Source: https://en.wikipedia.org/wiki/Greedoid?oldid=643814038 Contributors: Edward, Michael Hardy, Charles Matthews,Dcoetzee, Zoicon5, Altenmann, Peterkwok, Peter Kwok, Zaslav, Nickj, Oleg Alexandrov, Mathbot, Gaius Cornelius, Claygate, Grin-Bot~enwiki, Bluebot, Mhym, Dreadstar, E-Kartoffel, Sytelus, Headbomb, LachlanA, David Eppstein, LokiClock, Justin W Smith, HansAdler, Addbot, Citation bot, Citation bot 1, Kiefer.Wolfowitz, SporkBot, Desikblack, Nosuchforever, Deltahedron, Dillon128, JMP EAXand Anonymous: 11

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• Kruskal–Katona theorem Source: https://en.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem?oldid=672464638 Contribu-tors: TakuyaMurata, Charles Matthews, Psychonaut, Robinh, Giftlite, Jkseppan, Bender235, Gene Nygaard, BeteNoir, Bluebot, CBM,Sopoforic, Hermel, A3nm, David Eppstein, Geometry guy, Arcfrk, Addbot, Yobot, Jowa fan, ZéroBot, Brad7777 and Anonymous: 6

• Levi graph Source: https://en.wikipedia.org/wiki/Levi_graph?oldid=514913839 Contributors: Tomo, Infrogmation, Michael Hardy,Booyabazooka, AugPi, Charles Matthews, Alan Liefting, Giftlite, Rgdboer, Remuel, Raymond, Linas, Mathbot, AntiVandalBot, DavidEppstein, Koko90, JohnnyMrNinja, Jarauh, Justin W Smith, Twri, Citation bot 1 and Anonymous: 3

• Matroid Source: https://en.wikipedia.org/wiki/Matroid?oldid=678976987Contributors: AxelBoldt, Michael Hardy, Booyabazooka, Notheruser,Charles Matthews, Dcoetzee, Hbruhn, Dmytro, Altenmann, Robinh, Tobias Bergemann, Giftlite, Gubbubu, Alberto da Calvairate~enwiki,Almit39, Jqstm, Barnaby dawson, Rich Farmbrough, Zaslav, Gauge, Spayrard, Kwamikagami, EmilJ, Guidod, Pontus, Oleg Alexandrov,Natalya, Joriki, Rjwilmsi, Jweiss11, Salix alba, MZMcBride, R.e.b., VKokielov, Mathbot, Margosbot~enwiki, Bgwhite, Siddhant, Ency-clops, Michael Slone, Modify, SmackBot, RDBury, Tracy Hall, Papa November, Thomas Bliem, Mhym, Akriasas, Mdrine, Geminatea,

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• Pro-simplicial set Source: https://en.wikipedia.org/wiki/Pro-simplicial_set?oldid=611912045 Contributors: Michael Hardy, CharlesMatthews, Alphachimp, Kevinbrowning, CBM, Changbao, David Eppstein, Omnipaedista, Erik9bot, Davidaedwards and Anonymous: 1

• Property B Source: https://en.wikipedia.org/wiki/Property_B?oldid=678344414 Contributors: Schneelocke, Charles Matthews, Jnc,Blotwell, R.e.b., Eskimbot, Dreadstar, Cold Light, Rschwieb, King Bee, CharlotteWebb, David Eppstein, Cjwiki, Zuphilip, Addbot,Kilom691, Citation bot 1, RjwilmsiBot, Krenair, Dexbot, Vl.gusev, Anrnusna and Anonymous: 4

• Radon’s theorem Source: https://en.wikipedia.org/wiki/Radon’s_theorem?oldid=646088222 Contributors: Michael Hardy, CharlesMatthews, Tosha, Giftlite, Zaslav, Pouya, Sympleko, Rjwilmsi, Mathbot, Ewlyahoocom, YurikBot, Michael Slone, RDBury, BeteNoir,Reedy, CBM, Jikmo, Thijs!bot, Headbomb, David Eppstein, Val001, Synthebot, Addbot, Luckas-bot, Yobot, Kiefer.Wolfowitz, HelpfulPixie Bot, Brad7777 and Anonymous: 8

• Ring of sets Source: https://en.wikipedia.org/wiki/Ring_of_sets?oldid=654807888 Contributors: Michael Hardy, Charles Matthews,Chentianran~enwiki, Lethe, DemonThing, EmilJ, Touriste, Keenan Pepper, Salix alba, Trovatore, SmackBot, Keith111, David Eppstein,VolkovBot, Joeldl, DragonBot, PixelBot, Addbot, Jarble, Luckas-bot, Erik9bot, ZéroBot, Zephyrus Tavvier, Wlelsing, DarenCline andAnonymous: 8

• Sauer–Shelah lemma Source: https://en.wikipedia.org/wiki/Sauer%E2%80%93Shelah_lemma?oldid=681743017Contributors: MichaelHardy, Giftlite, Rjwilmsi, Eastmain, David Eppstein, Topher385, Yobot, SBaker43, HasteurBot and Anonymous: 2

• Segal space Source: https://en.wikipedia.org/wiki/Segal_space?oldid=610019953 Contributors: Michael Hardy, Rjwilmsi, R.e.b., Head-bomb, Bte99, Cavarrone, Hijigne and Anonymous: 1

• Set cover problem Source: https://en.wikipedia.org/wiki/Set_cover_problem?oldid=671504353 Contributors: Charles Matthews, Dcoet-zee, Kaal, Robinh, Mellum, Jason Quinn, Macrakis, TempeteCoeur, Zaslav, Cje~enwiki, JanKG, Chobot, Payool, SmackBot, Od Mishehu,J. Finkelstein, Ezrarez, Ylloh, Pgr94, Thijs!bot, Tatrgel, MrSampson, LokiClock, Singleheart, LiranKatzir, Keepssouth, Lisatwo, Svick,

266 CHAPTER 74. ∞-GROUPOID

Compupdate, Addbot, LatitudeBot, Yobot, Nallimbot, Citation bot, Winniehell, Miym, Omnipaedista, Yewang315, אורנים ,יוני Citationbot 1, Chenopodiaceous, RobinK, Deeparnab, EmausBot, AManWithNoPlan, Wcherowi, Rezabot, Jimmy-jambe, BG19bot, Qx2020,Aksh04ay, AustinBuchanan, Fabian Henneke, Ruimaranhao and Anonymous: 51

• Shapley–Folkman lemma Source: https://en.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma?oldid=675404570 Contribu-tors: Michael Hardy, Giftlite, Rich Farmbrough, Bender235, Cretog8, Rjwilmsi, Koavf, Sodin, Volunteer Marek, A. Pichler, Myasuda,Malleus Fatuorum, Headbomb, Lfstevens, David Eppstein, Geometry guy, Mild Bill Hiccup, Dank, Redhill54, Paulmnguyen, Eklipse,Addbot, Forich, Jarble, CountryBot, Yobot, AnomieBOT, Citation bot, DynamoDegsy, LilHelpa, Xqbot, Gilo1969, FrescoBot, Re-drose64, Kiefer.Wolfowitz, Jonesey95, Tomcat7, RjwilmsiBot, GA bot, GoingBatty, ZéroBot, Bpdmit, Helpful Pixie Bot, BattyBot,Dexbot, Monkbot and Anonymous: 5

• Sigma-algebra Source: https://en.wikipedia.org/wiki/Sigma-algebra?oldid=676073696Contributors: AxelBoldt, Zundark, Tarquin, Iwn-bap, Miguel~enwiki, Michael Hardy, Chinju, Karada, Stevan White, Charles Matthews, Dysprosia, Vrable, AndrewKepert, Fibonacci,Robbot, Romanm, Ruakh, Giftlite, Lethe, MathKnight, Mboverload, Gubbubu, Gauss, Barnaby dawson, Vivacissamamente, WilliamElliot, ArnoldReinhold, Paul August, Bender235, Zaslav, Elwikipedista~enwiki, MisterSheik, EmilJ, SgtThroat, Jung dalglish, Tsirel,Passw0rd, Msh210, Jheald, Cmapm, Ultramarine, Oleg Alexandrov, Linas, Graham87, Salix alba, FlaBot, Mathbot, Jrtayloriv, Chobot,Jayme, YurikBot, Lucinos~enwiki, Archelon, Trovatore, Mindthief, Solstag, Crasshopper, Dinno~enwiki, Nielses, SmackBot, Melchoir,JanusDC, Object01, Dan Hoey, MalafayaBot, RayAYang, Nbarth, DHN-bot~enwiki, Javalenok, Gala.martin, Henning Makholm, Lam-biam, Dbtfz, Jim.belk, Mets501, Stotr~enwiki, Madmath789, CRGreathouse, CBM, David Cooke, Mct mht, Blaisorblade, Xanthar-ius, , Thijs!bot, Escarbot, Keith111, Forgetfulfunctor, Quentar~enwiki, MSBOT, Magioladitis, RogierBrussee, Paartha, Joeabauer,Hippasus, Policron, Cerberus0, Digby Tantrum, Jmath666, Alfredo J. Herrera Lago, StevenJohnston, Ocsenave, Tcamps42, SieBot, Mel-combe, MicoFilós~enwiki, Andrewbt, The Thing That Should Not Be, Mild Bill Hiccup, BearMachine, 1ForTheMoney, DumZiBoT, Ad-dbot, Luckas-bot, Yobot, Li3939108, Amirmath, Godvjrt, Xqbot, RibotBOT, Charvest, FrescoBot, BrideOfKripkenstein, AstaBOTh15,Stpasha, RedBot, Soumyakundu, Wikiborg4711, Stj6, TjBot, Max139, KHamsun, Rafi5749, ClueBot NG, Thegamer 298, QuarkyPi,Brad7777, AntanO, Shalapaws, Crawfoal, Dexbot, Y256, Jochen Burghardt, A koyee314, Limit-theorem, Mark viking, NumSPDE,Moyaccercchi, Killaman slaughtermaster, DarenCline and Anonymous: 82

• Sigma-ideal Source: https://en.wikipedia.org/wiki/Sigma-ideal?oldid=607167568Contributors: Charles Matthews, RickK, Vivacissama-mente, Linas, Magister Mathematicae, RussBot, SmackBot, Valfontis, Yobot and Anonymous: 3

• Sigma-ring Source: https://en.wikipedia.org/wiki/Sigma-ring?oldid=667498132 Contributors: Charles Matthews, Touriste, Salix alba,Maxbaroi, Konradek, Keith111, Addbot, Luckas-bot, Xqbot, ZéroBot, Zephyrus Tavvier, Jim Sukwutput, Brirush, DarenCline, Gana-tuiyop and Anonymous: 3

• Simplex category Source: https://en.wikipedia.org/wiki/Simplex_category?oldid=605945486 Contributors: Michael Hardy, Silverfish,Charles Matthews, Kine, Oleg Alexandrov, Linas, R.e.b., Masnevets, Marc Harper, SmackBot, , David Eppstein, Malik Shabazz,LokiClock, Tcamps42, Wikidsp, SchreiberBike, Ajstern, ComputScientist, EmausBot, John of Reading and Anonymous: 7

• Simplicial approximation theorem Source: https://en.wikipedia.org/wiki/Simplicial_approximation_theorem?oldid=648548614 Con-tributors: Charles Matthews, Nonick, Giftlite, Oleg Alexandrov, Rjwilmsi, MZMcBride, SmackBot, BeteNoir, Reedy Bot, Geometry guy,Omaranto, Addbot, Lightbot, Luckas-bot, Erik9bot, Tonyxty, Brad7777, K9re11 and Anonymous: 3

• Simplicial complex Source: https://en.wikipedia.org/wiki/Simplicial_complex?oldid=677467215Contributors: Tomo, Charles Matthews,1984, Altenmann, Ashwin, Tosha, Giftlite, BenFrantzDale, Tomruen, TedPavlic, Zaslav, Gauge, Oleg Alexandrov, Staecker, FlaBot,YurikBot, Gene.arboit, Gadget850, Sardanaphalus, SmackBot, UU, Henning Makholm, Mathsci, Equendil, Cydebot, Michael Fourman,Thijs!bot, D Haggerty, .anacondabot, Magioladitis, David Eppstein, Smithers888, VolkovBot, Trevorgoodchild, Neparis, PixelBot, Ad-dbot, Luckas-bot, Cflm001, Af2125, Erel Segal, Drilnoth, Control.valve, Prijutme4ty, Undsoweiter, Molitorppd22, Citation bot 1, Jowafan, EmausBot, Zzzgggrrr, Frietjes, BTotaro, BG19bot, Brad7777, Samreid94, KasparBot and Anonymous: 23

• Simplicial group Source: https://en.wikipedia.org/wiki/Simplicial_group?oldid=576996426Contributors: TakuyaMurata, Bearcat, Dekartand Anonymous: 1

• Simplicial homotopy Source: https://en.wikipedia.org/wiki/Simplicial_homotopy?oldid=669911277Contributors: TakuyaMurata, Bearcat,David Eppstein, AnomieBOT and Dong, where is my automobile?

• Simplicialmanifold Source: https://en.wikipedia.org/wiki/Simplicial_manifold?oldid=676626694Contributors: Silverfish, Charles Matthews,Fropuff, C S, Alamino, Kevin Lamoreau, Blotwell, Anthony Appleyard, Oleg Alexandrov, R.e.b., JonathanD, SmackBot, Maksim-e~enwiki, UU, CBM, Giansira, Barticus88, David Eppstein, R'n'B, Neparis, 1ForTheMoney, AnomieBOT, LilHelpa, Point-set topologist,Erik9bot, Gofors, MerlIwBot, Michelino12, ChrisGualtieri and Anonymous: 3

• Simplicial presheaf Source: https://en.wikipedia.org/wiki/Simplicial_presheaf?oldid=653900557Contributors: TakuyaMurata, Jakob.scholbach,MenoBot II, SuperJew, John of Reading, ChrisGualtieri, Spectral sequence and Anonymous: 1

• Simplicial set Source: https://en.wikipedia.org/wiki/Simplicial_set?oldid=678511490Contributors: AxelBoldt, Michael Hardy, Alodyne,TakuyaMurata, Charles Matthews, Aenar, Giftlite, LockeShocke, Four, Gauge, Kine, Msh210, Linas, R.e.b., Marc Harper, SmackBot,UU, Tbjw, Sadalmelik, CBM, , Jakob.scholbach, David Eppstein, Willow1729, Alexey Muranov, Rswarbrick, Beroal, Addbot, Luckas-bot, Yobot, Citation bot, Freebirth Toad, Anne Bauval, Shadowjams, Citation bot 1, SuperJew, Quondum, Sahimrobot, Aban1313 andAnonymous: 28

• Sperner’s theorem Source: https://en.wikipedia.org/wiki/Sperner’s_theorem?oldid=621493807 Contributors: Michael Hardy, PeterKwok, Rjwilmsi, David Eppstein and ChrisGualtieri

• Steiner system Source: https://en.wikipedia.org/wiki/Steiner_system?oldid=681920463 Contributors: AxelBoldt, Tobias Hoevekamp,Zundark, Taw, Roadrunner, Karl Palmen, Chas zzz brown, Michael Hardy, Darkwind, Charles Matthews, Dcoetzee, Giftlite, Gene WardSmith, Fropuff, Sigfpe, Zaslav, Btyner, Rjwilmsi, Salix alba, R.e.b., John Baez, Ninel, Jemebius, Cullinane, Evilbu, SmackBot, RDBury,Nbarth, Cícero, Makyen, RJChapman, Cydebot, Vanish2, David Eppstein, JoergenB, Mkoko483, JackSchmidt, Sfan00 IMG, Justin WSmith, Evgumin, Watchduck, Addbot, Fractaler, Anybody, Citation bot 1, RjwilmsiBot, GoingBatty, ה ,.אריה Wcherowi, Helpful PixieBot, Deltahedron, Saung Tadashi and Anonymous: 34

• Symmetric spectrum Source: https://en.wikipedia.org/wiki/Symmetric_spectrum?oldid=587678902Contributors: TakuyaMurata, Rack-lever, Alvin Seville and Anonymous: 1

74.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 267

• Teichmüller–Tukey lemma Source: https://en.wikipedia.org/wiki/Teichm%C3%BCller%E2%80%93Tukey_lemma?oldid=649839310Contributors: Michael Hardy, ,דוד Giftlite, Paul August, Malcolma, Incnis Mrsi, CBM, Marek69, TreasuryTag, Fcady2007, Addbot,Yobot, Erik9bot, HRoestBot, FoxBot, John of Reading, Syberlazer, Brirush, Juanbenitez125 and Anonymous: 1

• Tverberg’s theorem Source: https://en.wikipedia.org/wiki/Tverberg’s_theorem?oldid=621918770Contributors: Michael Hardy, CharlesMatthews, Giftlite, Rjwilmsi, Rschwieb, David Eppstein, Luckas-bot, Kiefer.Wolfowitz, KLBot2 and Brad7777

• Two-graph Source: https://en.wikipedia.org/wiki/Two-graph?oldid=670009077 Contributors: Giftlite, Zaslav, Rjwilmsi, Ttzz, DavidEppstein, Tedyun, Yobot, Twri, RobinK, Wcherowi and Anonymous: 4

• Ultrafilter Source: https://en.wikipedia.org/wiki/Ultrafilter?oldid=679000534Contributors: AxelBoldt, Zundark, Michael Hardy, Chinju,Stevan White, Charles Matthews, Timwi, Prumpf, MathMartin, Giftlite, Gene Ward Smith, Markus Krötzsch, Jason Quinn, Salasks, PaulAugust, EmilJ, TenOfAllTrades, Oleg Alexandrov, Graham87, Rjwilmsi, R.e.b., Vlad Patryshev, FlaBot, Karelj, YurikBot, Benja, Trova-tore, Crasshopper, Arthur Rubin, Psolrzan, Eskimbot, Mhss, Henning Makholm, Physis, JRSpriggs, CRGreathouse, Gregbard, NickNumber, JAnDbot, .anacondabot, Magioladitis, EdwardLockhart, Sullivan.t.j, David Eppstein, Danimey, Quux0r, Gogobera, VolkovBot,Cbigorgne, M gol, Anchor Link Bot, Nsk92, Mpd1989, Hans Adler, Hugo Herbelin, Legobot, Luckas-bot, Yobot, Ht686rg90, Kilom691,AnomieBOT, Xqbot, Howard McCay, FrescoBot, Theorist2, Citation bot 1, Tkuvho, RjwilmsiBot, EmausBot, Wgunther, Bbbbbbbbba,Nosuchforever, CitationCleanerBot, Dexbot, Mark viking, Grabigail and Anonymous: 24

• Union-closed sets conjecture Source: https://en.wikipedia.org/wiki/Union-closed_sets_conjecture?oldid=641681074Contributors: CharlesMatthews, Markhurd, Giftlite, Trovatore, Bluebot, W109, David Eppstein, Addbot, Mjcollins68, Lightbot, Yobot, Ptrf, Trappist the monk,ZéroBot, Anuj1123 and Anonymous: 4

• Universal set Source: https://en.wikipedia.org/wiki/Universal_set?oldid=679200469 Contributors: Awaterl, Patrick, Charles Matthews,Dysprosia, Hyacinth, Paul August, Jumbuck, Gary, Salix alba, Chobot, Hairy Dude, SmackBot, Incnis Mrsi, FlashSheridan, Gilliam,Lambiam, AndriusKulikauskas, Newone, CBM, User6985, Cydebot, LookingGlass, David Eppstein, Ttwo, VolkovBot, Anonymous Dis-sident, SieBot, ToePeu.bot, Oxymoron83, Cliff, Addbot, Neodop, Download, Dimitris, Yobot, Shlakoblock, Citation bot, Xqbot, Amaury,FrescoBot, Aikidesigns, Petrb, Wcherowi, Jochen Burghardt, Vivianthayil, Smortypi, Blackbombchu, ColRad85, TerryAlex, Arian DMand Anonymous: 24

• Universe (mathematics) Source: https://en.wikipedia.org/wiki/Universe_(mathematics)?oldid=672270017Contributors: Mav, The Anome,Andre Engels, Toby Bartels, William Avery, Michael Hardy, Oliver Pereira, MartinHarper, Gabbe, Geoffrey~enwiki, Charles Matthews,Asar~enwiki, WhisperToMe, Jeoth, Robbot, Fredrik, Wile E. Heresiarch, Lethe, Waltpohl, Gdr, Discospinster, Rich Farmbrough, PaulAugust, Elwikipedista~enwiki, Mdd, Arthena, Linas, Isnow, Marudubshinki, Jshadias, Salix alba, John Baez, Mathbot, YurikBot, Zwobot,SmackBot, FlashSheridan, ArgentiumOutlaw, Acepectif, Lambiam, Scoty6776, Mets501, EdC~enwiki, Newone, CmdrObot, CBM,JAnDbot, Hut 8.5, Magioladitis, Zana Dark, David Eppstein, R'n'B, Huzzlet the bot, CooperDenn, Policron, Station1, Kumioko (re-named), Marino-slo, Hans Adler, DumZiBoT, Addbot, CarsracBot, Lightbot, Luckas-bot, Andy.melnikov, AnomieBOT, , XZeroBot,Erik9bot, Pinethicket, MastiBot, Level777, ClueBot NG, BG19bot, Kephir, AdhesiveStation and Anonymous: 28

• Vietoris–Rips complex Source: https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex?oldid=669231205Contributors: Ben-der235, Gaius Cornelius, Ranicki, David Eppstein, LokiClock, Subh83, LarRan, Sabalka, Yobot, Citation bot, Citation bot 1, Trappistthe monk, Helpful Pixie Bot, Cnoized, Lesser Cartographies and Anonymous: 2

• ∞-groupoid Source: https://en.wikipedia.org/wiki/%E2%88%9E-groupoid?oldid=648373461Contributors: TakuyaMurata, Octahedron80,Yobot, AnomieBOT, Alvin Seville, BG19bot and Anonymous: 2

74.3.2 Images• File:150614-PG-3-2-schoolgirls-arrangement.png Source: https://upload.wikimedia.org/wikipedia/commons/0/0a/150614-PG-3-2-schoolgirls-arrangement.

png License: CC BY-SA 4.0 Contributors: Own work Original artist: Steven H. Cullinane• File:2dKanFibration.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/ca/2dKanFibration.svg License: CC BY-SA 3.0Contributors: Own work Original artist: Rswarbrick

• File:Antimatroid.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/45/Antimatroid.svg License: Public domain Contrib-utors: Transferred from en.wikipedia to Commons. Original artist: David Eppstein at English Wikipedia

• File:Associatividadecat.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a7/Associatividadecat.svgLicense: Public do-main Contributors: This file was derived from Associatividadecat.png: <a href='//commons.wikimedia.org/wiki/File:Associatividadecat.png' class='image'><img alt='Associatividadecat.png' src='https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Associatividadecat.png/50px-Associatividadecat.png' width='50' height='57' srcset='https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Associatividadecat.png/75px-Associatividadecat.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Associatividadecat.png/100px-Associatividadecat.png 2x' data-file-width='253' data-file-height='287' /></a>Original artist: Associatividadecat.png: Campani

• File:BellNumberAnimated.gif Source: https://upload.wikimedia.org/wikipedia/commons/a/ab/BellNumberAnimated.gif License: CCBY-SA 3.0 Contributors: Own work Original artist: Xanthoxyl

• File:Caratheodorys_theorem_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/5a/Caratheodorys_theorem_example.svg License: BSD Contributors: Created by user:brighterorange, based on PNG by User:Dysprosia Original artist: Tom MurphyVII

• File:CardContin.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg License: Public domain Contrib-utors: en:Image:CardContin.png Original artist: en:User:Trovatore, recreated by User:Stannered

• File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Origi-nal artist: ?

• File:Complete-quads.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7b/Complete-quads.svg License: Public domainContributors: Transferred from en.wikipedia to Commons. Original artist: David Eppstein at English Wikipedia

• File:Convex_polygon_illustration1.png Source: https://upload.wikimedia.org/wikipedia/commons/0/06/Convex_polygon_illustration1.png License: Public domain Contributors: self-made, with Inkscape. Replaced an older version of the picture made by me also. Originalartist: Oleg Alexandrov

268 CHAPTER 74. ∞-GROUPOID

• File:Convex_polygon_illustration2.png Source: https://upload.wikimedia.org/wikipedia/commons/1/11/Convex_polygon_illustration2.png License: Public domain Contributors: self-made, with Inkscape. Replaced an older version of the picture made by me also. Originalartist: Oleg Alexandrov

• File:Convex_shelling.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/37/Convex_shelling.svg License: Public domainContributors: Transferred from en.wikipedia to Commons. Original artist: David Eppstein at English Wikipedia

• File:Disjuct-sets.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/89/Disjuct-sets.svg License: CC BY-SA 3.0 Contrib-utors: Own work Original artist: Svjo

• File:Disjunkte_Mengen.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Disjunkte_Mengen.svg License: CC BY3.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla)

• File:Double_six.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9a/Double_six.svg License: Public domain Contribu-tors: Own work Original artist: David Eppstein

• File:E-to-the-i-pi.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/E-to-the-i-pi.svg License: CC BY 2.5 Contribu-tors: ? Original artist: ?

• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: TheTango! Desktop Project. Original artist:The people from the Tango! project. And according to the meta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

• File:Epigraph_convex.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/31/Epigraph_convex.svg License: CC BY-SA3.0 Contributors: Own work Original artist: Eli Osherovich

• File:Extreme_points_illustration.png Source: https://upload.wikimedia.org/wikipedia/commons/c/c8/Extreme_points_illustration.pngLicense: Public domain Contributors: ? Original artist: ?

• File:Fano_Plane.jpg Source: https://upload.wikimedia.org/wikipedia/commons/5/57/Fano_Plane.jpg License: Public domain Contrib-utors: Own work Original artist: Polarbearcat

• File:Fano_plane-Levi_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/01/Fano_plane-Levi_graph.svgLicense:CC BY-SA 3.0 Contributors: Own work Original artist: Tremlin

• File:Fano_plane.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/af/Fano_plane.svg License: Public domain Contribu-tors: created using vi Original artist: de:User:Gunther

• File:Fano_plane_with_nimber_labels.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/32/Fano_plane_with_nimber_labels.svg License: Public domain Contributors:

• Fano_plane.svg Original artist: Fano_plane.svg: de:User:Gunther• File:GQ(2,2),_the_Doily.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a9/GQ%282%2C2%29%2C_the_Doily.svgLicense: CC BY-SA 3.0 Contributors: Own work Original artist: Wcherowi

• File:GQ24.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/59/GQ24.svg License: CC BY 3.0 Contributors: Own workOriginal artist: Rocchini

• File:Genji_chapter_symbols_groupings_of_5_elements.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/ac/Genji_chapter_symbols_groupings_of_5_elements.svg License: Public domain Contributors: Self-made graphic, generated from abstract geom-etry and publicly-available information. Own work Original artist: AnonMoos

• File:Helly’s_theorem.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/65/Helly%27s_theorem.svg License: Publicdomain Contributors: Own work Original artist: David Eppstein

• File:Incidencestructure.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2a/Incidencestructure.svg License: CC0 Con-tributors: Own work Original artist: Wcherowi

• File:Indiabundleware.jpg Source: https://upload.wikimedia.org/wikipedia/commons/7/79/Indiabundleware.jpg License: Public do-main Contributors: Own collection Original artist: scanned by User:Nickpo

• File:Indifference_curves_showing_budget_line.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ed/Indifference_curves_showing_budget_line.svg License: CC BY-SA 3.0 Contributors:

• Simple-indifference-curves.svg Original artist: Simple-indifference-curves.svg: SilverStar at en.wikipedia• File:Inner_radius.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8d/Inner_radius.svg License: Public domain Con-tributors: Own work Original artist: David Eppstein

• File:Intersecting_set_families_2-of-4.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/86/Intersecting_set_families_2-of-4.svg License: CC0 Contributors: Own work Original artist: David Eppstein

• File:Kan_fibration.png Source: https://upload.wikimedia.org/wikipedia/en/b/b4/Kan_fibration.png License: Cc-by-sa-3.0 Contribu-tors: ? Original artist: ?

• File:Kenneth_Arrow,_Stanford_University.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/9d/Kenneth_Arrow%2C_Stanford_University.jpg License: CC BY 3.0 Contributors: Stanford News Service Original artist: Linda A. Cicero / Stanford NewsService

• File:KleinBottle-01.png Source: https://upload.wikimedia.org/wikipedia/commons/4/46/KleinBottle-01.png License: Public domainContributors: ? Original artist: ?

• File:Kleinsche_Flasche.png Source: https://upload.wikimedia.org/wikipedia/commons/b/b9/Kleinsche_Flasche.png License: CC BY-SA 3.0 Contributors: Own work Original artist: Rutebir

• File:LGDiary_CoverAndPage.jpg Source: https://upload.wikimedia.org/wikipedia/commons/5/5f/LGDiary_CoverAndPage.jpg Li-cense: Public domain Contributors: The Lady’s and Gentleman’s Diary, 1850, p. 48 Original artist: Thomas Penyngton Kirkman 1806-1895

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• File:Mappings-as-moduli.png Source: https://upload.wikimedia.org/wikipedia/en/5/58/Mappings-as-moduli.png License: PD Con-tributors: ? Original artist: ?

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270 CHAPTER 74. ∞-GROUPOID

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