problems in geometric and topological combinatorics · 2012-09-03 · gil kalai fantasies in...

Problems in Geometric and TopologicalCombinatorics

Gil Kalai

Berlin, October 2011

Gil Kalai Fantasies in Geometric and Topological Combinatorics

This lecture

1. Around Tverberg’s Theorem

2. Borsuk’s problem and the combinatorics of cocycles

3. A remark about connectivity

4. The Fractional Helly Property and homology growth


I. Around Tverberg’s theorem


Tverberg’s theorem

Tverberg’s theorem: Let X = {x1, x2, . . . , xm} be a set of mpoints in Rd , m ≥ (d + 1)(r − 1) + 1. Then X can be partitionedinto r pairwise disjoint parts X1,X2 . . . , Xr such that

conv(X1) ∩ conv(X2) ∩ · · · ∩ conv(Xr ) 6= ∅.

History: Birch (conjectured), Rado (proved a weaker result),Tverberg (proved), Tverberg (reproved), Tverberg and Vrecica(reproved), Sarkaria (reproved), Roundeff (reproved) (The easycase r = 2 is Radon’s theorem.)


The Topological Tverberg’s Conjecture

Topological Tverberg’s Conjecture: Let f : ∆(d+1)(r−1) → Rd

be a continuous function from the (d + 1)(r − 1) dimensionalsimplex to Rd . Then there are r disjoint faces of the simplexwhose images have a point in common.

The topological Tverberg’s conjecture is known to hold when r is aprime power. History: Barany and Bajmoczy , Barany, Shlosmanand Szucs, ... Zivaljevic and Vrecica, Blagojevic, Matschke, andZiegler


The dimensions of Tverberg’s points

Let X be a set of points in Rd . The Tverberg points of order r ,denoted by Tr (X ), are those points that belong to the intersectionof the convex hulls of r pairwise disjoint subsets of X .

The Cascade Conjecture:

|X |∑i=1

dim Ti (X ) ≥ 0.


The dimensions of Tverberg’s points: a weaker conjecture

The Weak Cascade Conjecture:

|X |∑i=1

dim conv(Ti (X )) ≥ 0.


Why Tverberg’s conjecture follows

Let X be a set of m = (r − 1)(d + 1) + 1 points in Rd . Thendim Ti (X ) ≤ d , for every i . If Tr (X ) is empty then

m∑i=1

dim Ti (X ) ≤

(r − 1)d + (−1)((d + 1)(r − 1) + 1− (r − 1)) = −1.


An even weaker conjecture: the dimensions of the k-cores

Let X be a set of points in Rd . The rth core of X , denoted Cr (X ),is the set of all the points that belong to every convex hull of allbut r of the points.Tr (X ) ⊂ Cr (X ).

Conjecture:|X |∑i=1

dim Ci (X ) ≥ 0.

I think this should be doable.


An even weaker statement

Let X be a set of points in Rd . Denoted by Ar (X ), those pointsthat belong to the intersection of the affine hull of r pairwisedisjoint subsets of X .I think this is essentially known:

|X |∑i=1

(dim Ai (X )) ≥ 0.


Kadari’s theorem:

Theorem: (Kadari 81-90) The cascade conjecture holds in theplane.

Uses (to the best of my memory) a claim that in the plane Cr (X )is the convex hull of Tr (X ). (Not true for d ≥ 3.)


A new∗ approach∗∗ to topological Tverberg

(Old approach) Divide your set to 3 parts (works only if 3 is aprimes)(New Approach) Divide your set into two parts and divide one partagain into two parts.(Something that might be needed:) If the set of Radon’s partitionsis sufficiently “connected” then a Tverberg’s partition into threeparts exists.

∗ not new ∗∗ not quite an approach more like a fantasy


Boris Bukh disproved the partition conjecture!

Let G be a family of subsets of a ground set X which is closedunder intersection. Define tr (G) to be the smallest integer with thefollowing property: Every set of tr (G) points from X can be dividedinto r parts, X1,X2, . . . ,Xr such that for every S1,S2, . . . ,Sr ∈ Gwith Xi ⊂ Si there is a point in common to all the S ′

i s.

The partition conjecture (disproved by Boris Bukh):tr − 1 ≤ r(t2 − 1).

Question: Does Tverberg’s theorem hold for oriented matroids?


II. Borsuk’s problem and cocycles


Borsuk’s conjecture

Karol Borsuk conjectured in 1933 that every bounded set in Rd

can be covered by d + 1 sets of smaller diameter.

Let f (d) be the smallest integer such that every set of diameter 1in Rd can be covered by f (d) sets of smaller diameter.


Larman’s conjecture

David Larman proposed to consider purely combinatorial specialcases

Conjecture: Let F be a family of subsets of {1, 2, . . . , n}, andsuppose that the symmetric difference between every two sets in Fhas at most t elements. Then F can be divided into n + 1 familiessuch that the symmetric difference between any pair of sets in thesame family is at most t − 1.

To see the connection with Borsuk’s problem just consider the setof characteristic vectors of the sets in the family.


Another question by Larman

Problem: Does Borsuk’s conjecture hold for 2-distance sets?


The cut construction

The construction of Jeff Kahn and myself can (essentially) bedescribed as follows:

The cut construction: The ground set is the set of edges of thecomplete graph on 4p vertices. The family F consists of allsubsets of edges which represent the edge set of a completebipartite graph.

The cut constructions shows that f (d) > exp(K√

d). We wouldlike to replace d1/2 by a larger exponent.


Cocycles

Definition: A k-cocycle is a collection of (k + 1)-subsets such thatevery (k + 2)-set T contains an even number of sets in thecollection.An alternative definition is to start with a collection G of k-setsand consider all (k + 1)-sets that contain an odd number ofmembers in G.It is easy to see that the two definitions are equivalent. (Thisequivalence expresses the fact that the k-cohomology of a simplexis zero.) Note that the symmetric difference of two cocycles is acocycle. In other words, the set of k-cocycles form a subspace overZ/2Z, i.e., a linear binary code.


Cocycles (cont.)

Definition: A k-cocycle is a collection of (k + 1)-subsets such thatevery (k + 2)-set T contains an even number of sets in thecollection.

1-cocycles correspond to cuts in graphs. Those were studiedintensively in the combinatorics literature. 2-cocycles were studiedunder the name “two-graphs”. Their study was initiated by J. J.Seidel.


The combinatorics of cocycles

Problem: Let k be odd. What is the maximum number ofsimplices in a k-dimensional cocycle with n vertices?


The combinatorics of cocycles (cont.)

There are various interesting combinatorial questions aboutcocycles. Yuval Peled (graduate sudent) has some results.

Let e(k, n) be the number of k-cocycles on n vertices.

Lemma: Two collections of k-sets (in the second definition)generate the same k-cocycle if and only if their symmetricdifference is a (k − 1)-cocycle.

It follows that e(k, n) = 2(nk)/e(k − 1, n). So e(k, n) = 2(n−1

k ).


The proposed construction

Construction: Consider all k dimensional cocycles on n vertices.(regarded as families of (k + 1)-tuples.)


Frankl-Rodl conjecture for cocycles

Conjecture: For every α > 0 there is β > 0 such that thefollowing holds: Let m be an integer so that the number ofk-cocycles with n vertices is at least exp(αnk). If F is a family ofcocycles such that the symmetric difference of no two cocycles inF has precisely m (k + 1)-sets. Then

|F| ≤ 2(1−β)(nk).


[Interlude: A remark about connectivity]

In his lecture Anders Bjorner asked about a two-dimensional arrayof connectivity notions for two positive integers (d , k):Horizontally, when d = 1 we have the notions coming from graphtheory of k-connectivity of graphs, k = 1, 2, 3... . Vertically, whenk = 1 there are notions of d-dimensional connectivity of simplicialcomplexes based on homology. The question was to fill the table. I

mentioned a similar question when horizontally we have for graphsnotions related to infinitesimal rigidity for embeddings ink-dimensional space. (This is where I stopped.)


III. The fractional Helly property and homology growth


The fractional Helly property

Let F be a family of sets. F satisfies The weak fractional Hellyproperty (WFHP) with index k, if For every α there is β such thatfor every subfamily G of n sets if a fraction α of all k-subfamiliesare intersecting then a fraction β of all members of G havenonempty intersection.

The strong FHP with index k: Also α → 1 when β → 1.

Piercing property with index k: For every p > k there is f (p) suchthat if from every p sets k have a point in common there are f (p)points such that every set contains one of them.


Theorem (Katchalski and Liu, Eckhoff, Kalai): Convex sets in Rd

have the strong fractional Helly property with index d + 1.

Theorem (Alon and Kleitman): Convex sets in Rd have thepiercing property with index d + 1.

Theorem (Alon, Kalai, Matousek, Meshulam): Weak fractionalHelly implies piercing property with the same index.


The Barany-Matousek theorem

Integral Helly theorem: Let F be a collection of n convex sets inRd . If every 2d sets in F have an integer point in common thenthere is an integer point common to all of the sets.

Barany-Matousek Theorem:Sets of integer points in convex sets in Rd satisfy the weakfractional Helly property with index d + 1.

In particular:There is a positive constant α(d) such that the followingstatement holds:Let F be a collection of n convex sets in Rd . If every d + 1 sets inF have an integer point in common then there is an integer pointcommon to α(d)n of the sets.


The Barany-Matousek theorem

Integral Helly theorem: Let F be a collection of n convex sets inRd . If every 2d sets in F have an integer point in common thenthere is an integer point common to all of the sets.Barany-Matousek Theorem:Sets of integer points in convex sets in Rd satisfy the weakfractional Helly property with index d + 1.

In particular:There is a positive constant α(d) such that the followingstatement holds:Let F be a collection of n convex sets in Rd . If every d + 1 sets inF have an integer point in common then there is an integer pointcommon to α(d)n of the sets.


The Leray property

A simplicial complex is called d-Leray if all homology groups ofdimension d or more of all induced subcomplexes vanish.Examples:0-Leray = complete complexes1-Leray = chordal graphs

(immediate) d-Leray implies Helly number ≤ d + 1(hard) d-Leray implies (strong) fractional helly with index d + 1.


What type of properties implies (weak) fractional Helly?

Theorem: (Matousek) Bounded VC-dimension implies the weakfractional Helly property.


Complexes with polynomial homology growth

Definition: The total Betti number of a simplicial complex K isthe sum of all its Betti numbers.

Definition A hereditary class of simplicial complexes (a class closedunder induced subcomplexes) has polynomial homology growth ofindex k if there is a constant α so that every complex in the classwith m vertices has total Betti number bounded above by αmk .


Polynomial homology growth and the fractional Hellyproperty

Conjecture (Kalai and Meshulam): For a collection F of sets, theweak fractional Helly property of index k follows from polynomialgrowth of index k for the nerve.


The case k = 0

For a graph G , I (G ) is the independent complex of G and β(I (G ))is the sum of (reduced) Betti numbers of I (H).

Conjecture: Let G be a graph. If βI (H) < K for every inducedsubgraph then χ(G ) is bounded.

Maybe, maybe this is true even if beta is replaced by the (reduced)Euler characteristic χ.

What about K=1.Conjecture: β(I (H)) ≤ 1 for every induced subgraph H iff G doesnot contain an induced cycle of length 0(mod 3).Gyarfas type question: Is there a uniform upper bound for thechromatic number of all graphs G such that all induced cycles in Gare of length 1 or 2 modulo 3?


The case k = 0


Conjecture: Let G be a graph. If βI (H) < K for every inducedsubgraph then χ(G ) is bounded.Maybe, maybe this is true even if beta is replaced by the (reduced)Euler characteristic χ.



The case k = 0



What about K=1.Conjecture: β(I (H)) ≤ 1 for every induced subgraph H iff G doesnot contain an induced cycle of length 0(mod 3).

Gyarfas type question: Is there a uniform upper bound for thechromatic number of all graphs G such that all induced cycles in Gare of length 1 or 2 modulo 3?


The case k = 0





Bonus: Amenta’s theorem


Amenta’s theorem

Amenta’s theorem (1996): Let F be the family of union of rdisjoint compact convex sets in Rd . Then the Helly order of F is(d + 1)r .This was a conjecture of Grunbaum and Motzkin (1961).

Theorem (Alon-Kalai and Matousek): Let F be the family ofunion of r compact convex sets in Rd . Then the Helly order of Fis finite.


Topological Amenta

Theorem: (Kalai and Meshulam, 2008): Let F be the family ofunion of r disjoint contractible sets in Rd . Then the Helly order ofF is (d + 1)r .


Combinatorial Amenta

Theorem (Eckhoff and Nischke 2008): Let F be a family withHelly order k, let G consists of unions of at most r disjointmembers of F , then G has Helly order kr .


A fantastic extension of Helly’s theoren

Conjecture: Let F be the family of unions of two disjoint nonempty compact convex sets in Rd . Suppose that the intesection ofevery proper subfamily is also a union of two disjoint non emptyconvex sets. Then if |F| > d + 1 then the intersection of allmembers of F is non empty.


problems in geometric and topological combinatorics · 2012-09-03 · gil kalai fantasies in...

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