Set families k rFrom Wikipedia, the free encyclopedia

Contents

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

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

3 Antimatroid 73.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denition 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 Clique complex 225.1 Independence complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Flag complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Conformal hypergraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Combinatorial design 256.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Fundamental combinatorial designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 A wide assortment of other combinatorial designs . . . . . . . . . . . . . . . . . . . . . . . . . . 276.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Content (measure theory) 347.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.2 Integration of bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3 Duals of spaces of bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.4 Construction of a measure from a content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.4.1 Denition on open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.4.2 Denition on all sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.4.3 Construction of a measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Dedekind number 378.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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8.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.3 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.4 Summation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.5 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9 Delta-ring 419.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

10 Disjoint sets 4210.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

11 Dynkin system 4611.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12 ErdsKoRado theorem 4812.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.2 Families of maximum size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3 Maximal intersecting families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

13 Family of sets 5113.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14 Field of sets 5314.1 Fields of sets in the representation theory of Boolean algebras . . . . . . . . . . . . . . . . . . . . 53

14.1.1 Stone representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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14.1.2 Separative and compact elds of sets: towards Stone duality . . . . . . . . . . . . . . . . . 5314.2 Fields of sets with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

14.2.1 Sigma algebras and measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.2.2 Topological elds of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.2.3 Preorder elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.2.4 Complex algebras and elds of sets on relational structures . . . . . . . . . . . . . . . . . . 55

14.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

15 Finite character 5715.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

16 Finite intersection property 5816.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.5 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

17 Fishers inequality 6017.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

18 Generalized quadrangle 6218.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.5 Generalized quadrangles with lines of size 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.6 Classical generalized quadrangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.7 Non-classical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.8 Restrictions on parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

19 Greedoid 6619.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.2 Classes of greedoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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19.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.4 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

20 Helly family 6920.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.3 Helly dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.4 The Helly property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

21 Incidence structure 7221.1 Formal denition and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.3 Dual structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.4 Other terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

21.4.1 Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.4.2 Block designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

21.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.5.1 Incidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.5.2 Pictorial representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.5.3 Incidence graph (Levi graph) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

21.6 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7821.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7821.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7821.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7921.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

22 Kirkmans schoolgirl problem 8022.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.4 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

23 KruskalKatona theorem 8523.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

23.1.1 Statement for simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8523.1.2 Statement for uniform hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

23.2 Ingredients of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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23.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

24 Levi graph 8724.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8824.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

25 Matroid 8925.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

25.1.1 Independent sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.1.2 Bases and circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.1.3 Rank functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9025.1.4 Closure operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9025.1.5 Flats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9125.1.6 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

25.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9125.2.1 Uniform matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9125.2.2 Matroids from linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9225.2.3 Matroids from graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9425.2.4 Matroids from eld extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

25.3 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9525.3.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9525.3.2 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9525.3.3 Sums and unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

25.4 Additional terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9625.5 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

25.5.1 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9625.5.2 Matroid partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9725.5.3 Matroid intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9725.5.4 Matroid software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

25.6 Polynomial invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9725.6.1 Characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9725.6.2 Tutte polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

25.7 Innite matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925.9 Researchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10225.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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26 Maximum coverage problem 10426.1 ILP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10426.2 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10426.3 Known extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10526.4 Weighted version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10526.5 Budgeted maximum coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10526.6 Generalized maximum coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

26.6.1 Generalized maximum coverage algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 10626.7 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10626.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10626.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

27 Monotone class theorem 10727.1 Denition of a monotone class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.2 Monotone class theorem for sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

27.2.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.3 Monotone class theorem for functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

27.3.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.3.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

27.4 Results and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

28 Near polygon 10928.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11028.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11028.3 Regular near polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11028.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11128.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11128.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

29 Noncrossing partition 11229.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11229.2 Lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11229.3 Role in free probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11229.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

30 Partition of a set 11630.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11730.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11730.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11730.4 Renement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11830.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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30.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11830.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

31 Partition regularity 12431.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12431.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

32 Pi system 12632.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12632.2 Relationship to -Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

32.2.1 The - Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12732.3 -Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

32.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

32.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12932.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

33 Polar space 13033.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13033.2 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13033.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

34 Property B 13134.1 Values of m(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13134.2 Asymptotics of m(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13134.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

35 Ring of sets 13335.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13335.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

36 SauerShelah lemma 13536.1 Denitions and statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13636.2 The number of shattered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13636.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13636.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13636.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

37 Set cover problem 138

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37.1 Integer linear program formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.2 Hitting set formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.3 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.4 Low-frequency systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.5 Inapproximability results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.6 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

38 Sigma-algebra 14238.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

38.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14238.1.2 Limits of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14338.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

38.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14538.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14538.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

38.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14638.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14638.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

38.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14638.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 14638.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14638.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14738.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 14838.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 148

38.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14838.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14938.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

39 Sigma-ideal 15039.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

40 Sigma-ring 15140.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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40.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15140.3 Similar concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15140.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15140.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15240.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

41 Sperners theorem 15341.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15341.2 Partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15341.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15341.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

41.4.1 No long chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.4.2 p-compositions of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.4.3 No long chains in p-compositions of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.4.4 Projective geometry analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15541.4.5 No long chains in p-compositions of a projective space . . . . . . . . . . . . . . . . . . . . 155

41.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15541.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

42 Steiner system 15742.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

42.1.1 Finite projective planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842.1.2 Finite ane planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

42.2 Classical Steiner systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842.2.1 Steiner triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842.2.2 Steiner quadruple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842.2.3 Steiner quintuple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

42.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15942.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15942.5 Mathieu groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15942.6 The Steiner system S(5, 6, 12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

42.6.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16042.7 The Steiner system S(5, 8, 24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

42.7.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16142.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16142.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

43 TeichmllerTukey lemma 16443.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16443.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

CONTENTS xi

43.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16443.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

44 Two-graph 16544.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16544.2 Switching and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16544.3 Adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16744.4 Equiangular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16744.5 Strongly regular graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16744.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16744.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

45 Ultralter 16945.1 Formal denition for ultralter on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16945.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17045.3 Generalization to partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17045.4 Special case: Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17045.5 Types and existence of ultralters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17045.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17145.7 Ordering on ultralters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17245.8 Ultralters on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17245.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17245.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17245.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

46 Union-closed sets conjecture 17446.1 Equivalent forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17446.2 Families known to satisfy the conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17446.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17546.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17546.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17546.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

47 Universal set 17647.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

47.1.1 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17647.1.2 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

47.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17647.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17747.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

47.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17747.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17747.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

xii CONTENTS

48 Universe (mathematics) 17948.1 In a specic context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17948.2 In ordinary mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17948.3 In set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18048.4 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18148.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18248.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18248.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18248.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18248.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 183

48.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18348.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18648.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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 nite 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 DenitionsA family of non-empty nite subsets of a universal set S is an abstract simplicial complex if, for every set X in, and every non-empty subset Y X, Y also belongs to .The nite sets that belong to are called faces of the complex, and a face Y is said to belong to another face X if Y X, so the denition of an abstract simplicial complex can be restated as saying that every face of a face of a complex is itself a face of . The vertex set of is dened as V() = , the union of all faces of . The elements of thevertex set are called the vertices of the complex. So for every vertex v of , the set {v} is a face of the complex.The maximal faces of (i.e., faces that are not subsets of any other faces) are called facets of the complex. Thedimension of a face X in is dened 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() is dened as thelargest dimension of any of its faces, or innity if there is no nite bound on the dimension of the faces.The complex is said to be nite if it has nitely many faces, or equivalently if its vertex set is nite. Also, issaid to be pure if it is nite-dimensional (but not necessarily nite) and every facet has the same dimension. In otherwords, is pure if dim() is nite and every face is contained in a facet of dimension dim().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 is a simplicial complex L such that every face of L belongs to ; that is, L and L is asimplicial complex. A subcomplex that consists of all of the subsets of a single face of is often called a simplexof . (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 is the subcomplex of consisting of all of the faces of that have dimension at most d. Inparticular, the 1-skeleton is called the underlying graph of . The 0-skeleton of can be identied 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 , often denoted /Y or lk(Y), is the subcomplex of dened by

/Y := fX 2 j X \ Y = ?; X [ Y 2 g:Note that the link of the empty set is itself.

1

2 CHAPTER 1. ABSTRACT SIMPLICIAL COMPLEX

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

1.2 Geometric realizationWe 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, dene |K | as a subset of [0, 1]S consisting of functions t : S [0, 1] satisfying the two conditions:

Xs2S

ts = 1

fs 2 S : ts > 0g 2 Now think of [0, 1]S as the direct limit of [0, 1]A where A ranges over nite 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 dene a functor F from K to the category of topological spaces asfollows. For any face X K of dimension n, let F(X) = n be the standard n-simplex. The order on the vertex setthen species a unique bijection between the elements of X and vertices of n, ordered in the usual way e0 < e1 < ...< en. If Y X is a face of dimension m < n, then this bijection species a unique m-dimensional face of n. DeneF(Y) F(X) to be the unique ane linear embedding of m as that distinguished face of n, such that the map onvertices is order preserving.We can then dene the geometric realization |K | as the colimit of the functor F. More specically |K | is the quotientspace of the disjoint union

aX2K

F (X)

by the equivalence relation which identies a point y F(Y) with its image under the map F(Y) F(X), for everyinclusion Y X.If K is nite, then we can describe |K | more simply. Choose an embedding of the vertex set of K as an anelyindependent subset of some Euclidean space RN of suciently high dimension N. Then any face X K can beidentied 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 identied with n.

1.3 Examples As an example, let V be a nite subset of S of cardinality n + 1 and let K be the power set of V. Then K is calleda 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 ag 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 nite chains.Its homology groups and other topological invariants contain important information about the poset.

The VietorisRips complex is dened from any metric spaceM and distance by forming a simplex for everynite subset of M with diameter at most . It has applications in homology theory, hyperbolic groups, imageprocessing, and mobile ad hoc networking. It is another example of a ag complex.

1.4. ENUMERATION 3

1.4 EnumerationThe number of abstract simplicial complexes on n elements is one less than the nth Dedekind number. These numbersgrow 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).

1.5 See also KruskalKatona theorem

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

[2] Korte, Bernhard; Lovsz, Lszl; 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; dierent denitions ofsmall will result in dierent denitions of almost disjoint.

2.1 DenitionThemost common choice is to take small to mean nite. In this case, two sets are almost disjoint if their intersectionis nite, i.e. if

jA \Bj

6 CHAPTER 2. ALMOST DISJOINT SETS

2.2 Other meaningsSometimes almost disjoint is used in some other sense, or in the sense of measure theory or topological category.Here are some alternative denitions of almost disjoint that are sometimes used (similar denitions apply to innitecollections):

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

jA \Bj < :

The case of = 1 is simply the denition of disjoint sets; the case of

= @0

is simply the denition of almost disjoint given above, where the intersection of A and B is nite.

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}

abcaba

acacbccacabcbcba

{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 dierent sequences in which elements may be included.Dilworth (1940) was the rst 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 dening antimatroids as set systems are very similar to those ofmatroids, but whereasmatroids are denedby an exchange axiom (e.g., the basis exchange, or independent set exchange axioms), antimatroids are dened 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 articial intelligence, and the states of knowledge of human learners.

3.1 DenitionsAn antimatroid can be dened as a nite 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 denition as a formal language, that is, as a set of strings dened from a nitealphabet of symbols. A language L dening 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 prex 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 dened 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 prex of s is feasible, then L denes an antimatroid. Thus, thesetwo denitions lead to mathematically equivalent classes of objects.[2]

3.2 Examples A chain antimatroid has as its formal language the prexes of a single word, and as its feasible sets the setsof symbols in these prexes. For instance the chain antimatroid dened by the word abcd has as its formallanguage the strings {, a, ab, abc, abcd"} and as its feasible sets the sets , {a}, {a,b}, {a,b,c}, and{a,b,c,d}.[3]

A poset antimatroid has as its feasible sets the lower sets of a nite partially ordered set. By Birkhos 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 nite 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 prexes 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 nd 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 wemay 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 dened from B as the set of prexes of words in B. It is often convenientto dene antimatroids from basic words in this way, but it is not straightforward to write an axiomatic denition of

10 CHAPTER 3. ANTIMATROID

antimatroids in terms of their basic words.

3.4 Convex geometriesSee also: Convex set, Convex geometry and Closure operator

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

G = fU n S j S 2 Fg

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 dene antimatroids, the set system dening 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 dened in terms of a closure operator that maps any subset of U to its minimalclosed superset. To be a closure operator, should have the following properties:

() = : the closure of the empty set is empty. Any set S is a subset of (S). If S is a subset of T, then (S) must be a subset of (T). For any set S, (S) = ((S)).

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

If neither y nor z belong to (S), but z belongs to (S {y}), then y does not belong to (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 (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 latticesAny 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 nite distributive lattices, and can be characterized in several dierent 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 dened 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 nite 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, wemay refer to a lattice with any of these three properties as join-distributive.Any antimatroid gives rise to a nite join-distributive lattice, and any nite join-distributive lattice comes from anantimatroid in this way.[6] Another equivalent characterization of nite 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 nite 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 nite join-distributive lattice as an accessible family of sets closed under unions (that is, asan antimatroid) may be viewed as an analogue of Birkhos representation theorem under which any nite distributivelattice has a representation as a family of sets closed under unions and intersections.

3.6 Supersolvable antimatroidsMotivated by a problem of dening partial orders on the elements of a Coxeter group, Armstrong (2007) studied an-timatroids which are also supersolvable lattices. A supersolvable antimatroid is dened 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 dierence 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 dimensionIf 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 = fS [ T j S 2 A ^ T 2 Bg:

Note that this is a dierent operation than the join considered in the lattice-theoretic characterizations of antimatroids:it combines 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 prexes 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 Dilworths 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 prex 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 verydierent: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.

3.8 EnumerationThe 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 ApplicationsBoth 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 articial 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 dening 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 rst 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 eciently 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): 149, doi:10.1016/S0001-8708(02)00011-7.

Armstrong, Drew (2007), The sorting order on a Coxeter group, arXiv:0712.1047. Birkho, Garrett; Bennett, M.K. (1985), The convexity lattice of a poset,Order 2 (3): 223242, doi:10.1007/BF00333128. Bjrner, Anders; Ziegler, Gnter M. (1992), 8 Introduction to greedoids, in White, Neil, Matroid Appli-cations, Encyclopedia of Mathematics and its Applications 40, Cambridge: Cambridge University Press, pp.284357, 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): 197205, 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): 303317, doi:10.1016/S0304-3975(03)00221-4.

Dilworth, Robert P. (1940), Lattices with unique irreducible decompositions, Annals of Mathematics 41 (4):771777, 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): 290299, doi:10.1007/BF02482912.

Edelman, Paul H.; Saks, Michael E. (1988), Combinatorial representation and convex dimension of convexgeometries, Order 5 (1): 2332, 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 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. 535544, MR 608454.

Korte, Bernhard; Lovsz, Lszl; Schrader, Rainer (1991), Greedoids, Springer-Verlag, pp. 1943, ISBN3-540-18190-3.

Monjardet, Bernard (1985), A use for frequently rediscovering a concept,Order 1 (4): 415417, doi:10.1007/BF00582748. Parmar, Aarati (2003), Some Mathematical Structures Underlying Ecient Planning, AAAI Spring Sympo-sium on Logical Formalization of Commonsense Reasoning (PDF).

Chapter 4

Block design

This article is about block designs with xed 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, nite 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 Denition of a BIBD (or 2-design)Given a nite set X (of elements called points) and integers k, r, 1, we dene 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 of blocks containing givendistinct points x and y in X is also independent of the choices.Family in the above denition 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 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, )-design or a (v, b, r, k, )-design. The parameters are not all independent; v, k, and determine b and r, and not all combinations of v, k, and are possible. The two basic equations connecting theseparameters are

bk = vr;

(v 1) = r(k 1):These conditions are not sucient as, for example, a (43,7,1)-design does not exist.[3]

The order of a 2-design is dened to be n = r . 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, = + b 2r. A 2-design and its complement have the same order.A fundamental theorem, Fishers inequality, named after the statistician Ronald Fisher, is that b v in any 2-design.

14

4.2. SYMMETRIC BIBDS 15

4.2 Symmetric BIBDsThe case of equality in Fishers 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 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 points in common, then (X, B) is a symmetric block design.[6]

The parameters of a symmetric design satisfy

(v 1) = k(k 1):

This imposes strong restrictions on v, so the number of points is far from arbitrary. The BruckRyserChowlatheorem gives necessary, but not sucient, 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 planesMain article: Projective plane

Finite projective planes are symmetric 2-designs with = 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 = 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 innite family (with respect to having a constant value) of symmetric block designs.[7]

4.2.2 BiplanesA biplane or biplane geometry is a symmetric 2-design with = 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 nite 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 eld 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 ve 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-designsAn 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 rst row and rst 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 rst row and rst column and convert every1 to a 0. The resulting 01 matrixM 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-designsA 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,) 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 nite ane planes. A solution of the famous 15 schoolgirl problem is aresolution of a 2-(15,3,1) design.[17]

4.4 Generalization: t-designsGiven 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 blocks. The numbers v (the numberof elements of X), b (the number of blocks), k, r, , and t are the parameters of the design. The design may be calleda t-(v,k,)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosenarbitrarily. The equations are

i =

v it i

k it i

for i = 0; 1; : : : ; t;

4.5. STEINER SYSTEMS 17

where i is the number of blocks that contain any i-element set of points.Theorem:[18] Any t-(v,k,)-design is also an s-(v,k,)-design for any s with 1 s t. (Note that the lambda valuechanges 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-designsLet D = (X, B) be a t-(v,k,) 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, ) design. Note that derived designswith respect to dierent 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,) 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,) design, is extendable, then one of the following holds:

1. D is an Hadamard 2-design,2. v = ( + 2)(2 + 4 + 2), k = 2 + 3 + 1,3. v = 495, k = 39, = 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 ane plane, i.e., a 3-(n2 + 1, n + 1, 1) design, is called a niteinversive plane, or Mbius 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 SuzukiTits 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 systemsMain article: Steiner system

18 CHAPTER 4. BLOCK DESIGN

A Steiner system (named after Jakob Steiner) is a t-design with = 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 denition is relatively modern, generalizing the classical denition 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 denition, this naming system is no longer strictlyadhered to.Projective planes and ane planes are examples of Steiner systems under the current denition while only the Fanoplane (projective plane of order 2) would have been a Steiner system under the older denition.

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 R are