Problems in Geometric and Topological Combinatorics Gil Kalai - - PowerPoint PPT Presentation

Problems in Geometric and Topological Combinatorics

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

Gil Kalai Fantasies in Geometric and Topological Combinatorics

• I. Around Tverberg’s theorem

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Tverberg’s theorem Tverberg’s theorem: Let X = {x1, x2, . . . , xm} be a set of m points in Rd, m ≥ (d + 1)(r − 1) + 1. Then X can be partitioned into r pairwise disjoint parts X1, X2 . . . , Xr such that conv(X1) ∩ conv(X2) ∩ · · · ∩ conv(Xr) = ∅. History: Birch (conjectured), Rado (proved a weaker result), Tverberg (proved), Tverberg (reproved), Tverberg and Vrecica (reproved), Sarkaria (reproved), Roundeff (reproved) (The easy case r = 2 is Radon’s theorem.)

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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) dimensional simplex to Rd. Then there are r disjoint faces of the simplex whose images have a point in common. The topological Tverberg’s conjecture is known to hold when r is a prime power. History: B´ ar´ any and Bajm´

• czy , B´

ar´ any, Shlosman and Sz¨ ucs, ... Zivaljevic and Vrecica, Blagojevi´ c, Matschke, and Ziegler

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 intersection

• f the convex hulls of r pairwise disjoint subsets of X.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 intersection

• f the convex hulls of r pairwise disjoint subsets of X.

The Cascade Conjecture:

|X|

• i=1

dim Ti(X) ≥ 0.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

The dimensions of Tverberg’s points: a weaker conjecture The Weak Cascade Conjecture:

|X|

• i=1

dim conv(Ti(X)) ≥ 0.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Why Tverberg’s conjecture follows Let X be a set of m = (r − 1)(d + 1) + 1 points in Rd. Then dim 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.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 all but r of the points. Tr(X) ⊂ Cr(X). Conjecture:

|X|

• i=1

dim Ci(X) ≥ 0. I think this should be doable.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

An even weaker statement Let X be a set of points in Rd. Denoted by Ar(X), those points that belong to the intersection of the affine hull of r pairwise disjoint subsets of X. I think this is essentially known:

|X|

• i=1

(dim Ai(X)) ≥ 0.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Kadari’s theorem: Theorem: (Kadari 81-90) The cascade conjecture holds in the plane. 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.)

Gil Kalai Fantasies in Geometric and Topological Combinatorics

A new∗ approach∗∗ to topological Tverberg (Old approach) Divide your set to 3 parts (works only if 3 is a primes) (New Approach) Divide your set into two parts and divide one part again into two parts. (Something that might be needed:) If the set of Radon’s partitions is sufficiently “connected” then a Tverberg’s partition into three parts exists.

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

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Boris Bukh disproved the partition conjecture! Let G be a family of subsets of a ground set X which is closed under intersection. Define tr(G) to be the smallest integer with the following property: Every set of tr(G) points from X can be divided into r parts, X1, X2, . . . , Xr such that for every S1, S2, . . . , Sr ∈ G with Xi ⊂ Si there is a point in common to all the S′

i s.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Boris Bukh disproved the partition conjecture! Let G be a family of subsets of a ground set X which is closed under intersection. Define tr(G) to be the smallest integer with the following property: Every set of tr(G) points from X can be divided into r parts, X1, X2, . . . , Xr such that for every S1, S2, . . . , Sr ∈ G with 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).

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Boris Bukh disproved the partition conjecture! Let G be a family of subsets of a ground set X which is closed under intersection. Define tr(G) to be the smallest integer with the following property: Every set of tr(G) points from X can be divided into r parts, X1, X2, . . . , Xr such that for every S1, S2, . . . , Sr ∈ G with 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?

Gil Kalai Fantasies in Geometric and Topological Combinatorics

• II. Borsuk’s problem and cocycles

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 1 in Rd can be covered by f (d) sets of smaller diameter.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Larman’s conjecture David Larman proposed to consider purely combinatorial special cases Conjecture: Let F be a family of subsets of {1, 2, . . . , n}, and suppose that the symmetric difference between every two sets in F has at most t elements. Then F can be divided into n + 1 families such that the symmetric difference between any pair of sets in the same family is at most t − 1. To see the connection with Borsuk’s problem just consider the set

• f characteristic vectors of the sets in the family.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Another question by Larman Problem: Does Borsuk’s conjecture hold for 2-distance sets?

Gil Kalai Fantasies in Geometric and Topological Combinatorics

The cut construction The construction of Jeff Kahn and myself can (essentially) be described as follows: The cut construction: The ground set is the set of edges of the complete graph on 4p vertices. The family F consists of all subsets of edges which represent the edge set of a complete bipartite graph. The cut constructions shows that f (d) > exp(K √ d). We would like to replace d1/2 by a larger exponent.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Cocycles Definition: A k-cocycle is a collection of (k + 1)-subsets such that every (k + 2)-set T contains an even number of sets in the collection. An alternative definition is to start with a collection G of k-sets and consider all (k + 1)-sets that contain an odd number of members in G. It is easy to see that the two definitions are equivalent. (This equivalence expresses the fact that the k-cohomology of a simplex is zero.) Note that the symmetric difference of two cocycles is a

• cocycle. In other words, the set of k-cocycles form a subspace over

Z/2Z, i.e., a linear binary code.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Cocycles (cont.) Definition: A k-cocycle is a collection of (k + 1)-subsets such that every (k + 2)-set T contains an even number of sets in the collection. 1-cocycles correspond to cuts in graphs. Those were studied intensively in the combinatorics literature. 2-cocycles were studied under the name “two-graphs”. Their study was initiated by J. J. Seidel.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

The combinatorics of cocycles Problem: Let k be odd. What is the maximum number of simplices in a k-dimensional cocycle with n vertices?

Gil Kalai Fantasies in Geometric and Topological Combinatorics

The combinatorics of cocycles (cont.) There are various interesting combinatorial questions about

• cocycles. 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 symmetric difference is a (k − 1)-cocycle. It follows that e(k, n) = 2(n

k)/e(k − 1, n). So e(k, n) = 2(n−1 k ). Gil Kalai Fantasies in Geometric and Topological Combinatorics

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

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Frankl-Rodl conjecture for cocycles Conjecture: For every α > 0 there is β > 0 such that the following holds: Let m be an integer so that the number of k-cocycles with n vertices is at least exp(αnk). If F is a family of cocycles such that the symmetric difference of no two cocycles in F has precisely m (k + 1)-sets. Then |F| ≤ 2(1−β)(n

k). Gil Kalai Fantasies in Geometric and Topological Combinatorics

[Interlude: A remark about connectivity] In his lecture Anders Bj¨

• rner asked about a two-dimensional array
• f connectivity notions for two positive integers (d, k):

Horizontally, when d = 1 we have the notions coming from graph theory of k-connectivity of graphs, k = 1, 2, 3... . Vertically, when k = 1 there are notions of d-dimensional connectivity of simplicial complexes based on homology. The question was to fill the table. I mentioned a similar question when horizontally we have for graphs notions related to infinitesimal rigidity for embeddings in k-dimensional space. (This is where I stopped.)

Gil Kalai Fantasies in Geometric and Topological Combinatorics

• III. The fractional Helly property and homology growth

Gil Kalai Fantasies in Geometric and Topological Combinatorics

The fractional Helly property Let F be a family of sets. F satisfies The weak fractional Helly property (WFHP) with index k, if For every α there is β such that for every subfamily G of n sets if a fraction α of all k-subfamilies are intersecting then a fraction β of all members of G have nonempty 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) such that if from every p sets k have a point in common there are f (p) points such that every set contains one of them.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 the piercing property with index d + 1. Theorem (Alon, Kalai, Matousek, Meshulam): Weak fractional Helly implies piercing property with the same index.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

The Barany-Matousek theorem Integral Helly theorem: Let F be a collection of n convex sets in

• Rd. If every 2d sets in F have an integer point in common then

there is an integer point common to all of the sets.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

The Barany-Matousek theorem Integral Helly theorem: Let F be a collection of n convex sets in

• Rd. If every 2d sets in F have an integer point in common then

there is an integer point common to all of the sets. Barany-Matousek Theorem: Sets of integer points in convex sets in Rd satisfy the weak fractional Helly property with index d + 1. In particular: There is a positive constant α(d) such that the following statement holds: Let F be a collection of n convex sets in Rd. If every d + 1 sets in F have an integer point in common then there is an integer point common to α(d)n of the sets.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

The Leray property A simplicial complex is called d-Leray if all homology groups of dimension d or more of all induced subcomplexes vanish. Examples: 0-Leray = complete complexes 1-Leray = chordal graphs (immediate) d-Leray implies Helly number ≤ d + 1 (hard) d-Leray implies (strong) fractional helly with index d + 1.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

What type of properties implies (weak) fractional Helly? Theorem: (Matousek) Bounded VC-dimension implies the weak fractional Helly property.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Complexes with polynomial homology growth Definition: The total Betti number of a simplicial complex K is the sum of all its Betti numbers. Definition A hereditary class of simplicial complexes (a class closed under induced subcomplexes) has polynomial homology growth of index k if there is a constant α so that every complex in the class with m vertices has total Betti number bounded above by αmk.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Polynomial homology growth and the fractional Helly property Conjecture (Kalai and Meshulam): For a collection F of sets, the weak fractional Helly property of index k follows from polynomial growth of index k for the nerve.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 induced subgraph then χ(G) is bounded.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 induced subgraph then χ(G) is bounded. Maybe, maybe this is true even if beta is replaced by the (reduced) Euler characteristic χ.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 induced subgraph 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 does not contain an induced cycle of length 0(mod 3).

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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 induced subgraph 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 does not contain an induced cycle of length 0(mod 3). Gy´ arf´ as type question: Is there a uniform upper bound for the chromatic number of all graphs G such that all induced cycles in G are of length 1 or 2 modulo 3?

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Bonus: Amenta’s theorem

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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

Gil Kalai Fantasies in Geometric and Topological Combinatorics

Amenta’s theorem Amenta’s theorem (1996): Let F be the family of union of r disjoint 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 of union of r compact convex sets in Rd. Then the Helly order of F is finite.

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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

Gil Kalai Fantasies in Geometric and Topological Combinatorics

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

Gil Kalai Fantasies in Geometric and Topological Combinatorics

A fantastic extension of Helly’s theoren Conjecture: Let F be the family of unions of two disjoint non empty compact convex sets in Rd. Suppose that the intesection of every proper subfamily is also a union of two disjoint non empty convex sets. Then if |F| > d + 1 then the intersection of all members of F is non empty.

Gil Kalai Fantasies in Geometric and Topological Combinatorics