The Combinatorics of Cocycles and Borsuk’s Problem.


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.

Alternative definition: Start with a collection \cal G of k-sets and consider all (k+1)-sets that contain an odd number of members in \cal 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.

1-cocycles correspond to the set of edges of a complete bipartite graph. (Or, in other words, to cuts in the complete graphs.) The number of edges of a complete bipartite graph on n vertices is of the form k(n-k). There are 2^{n-1} 1-cocycles on n vertices altogether, and there are n \choose k cocycles with k(n-k) edges.

2-cocycles were studied under the name “two-graphs”. Their study was initiated by J. J. Seidel.

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

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}\choose {k}}/e(k-1,n). So e(k,n)= 2^{{n-1} \choose {k}}.

A very basic question is:

Problem 1: For k odd what is the maximum number f(k,n) of (k+1)-sets  of a k-cocycle with n vertices?

When k is even, the set of all (k+1)-subsets of {1,2,…,n} is a cocycle.

Problem 2: What is the value of m such that the number ef(k,n,m) of k-cocycles with n vertices and m k-sets is maximum?

When k is even the complement of a cocycle is a cocycle and hence ef(k,n,m)=ef(k,n,{{n}\choose{k+1}}-m). It is likely that in this case ef(k,n,m) is a unimodal sequence (apart from zeroes), but I don’t know if this is known. When k is odd it is quite possible that (again, ignoring zero entries) ef(n,m) is unimodal attaining its maximum when m=1/2 {{n} \choose {k+1}}.

Borsuk’s conjecture, Larman’s conjecture and bipartite graphs

Karol Borsuk conjectured in 1933 that every bounded set in R^d can be covered by d+1 sets of smaller diameter. David Larman proposed a purely combinatorial special case (that looked much less correct than the full conjecture.)

Larman’s conjecture: Let \cal F be an $latex r$-intersecting  family of k-subsets of \{1,2,\dots, n\}, namely \cal F has the property that every two sets in the family have at least r elements in common. Then $\cal F$ can be divided into n (r+1)-intersecting families.

Larman’s conjecture is a special case of Borsuk’s conjecture: Just consider the set of characteristic vectors of the sets in the family (and note that they all belong to a hyperplane.) The case r=1 of Larman’s conjecture is open and very interesting.

A slightly more general case of Borsuk’s conjecture is for sets of 0-1 vectors (or equivalently \pm 1 vectors. Again you can consider the question in terms of covering a family of sets by subfamilies. Instead of intersection we should consider symmetric differences.

Borsuk 0-1 conjecture: Let \cal F be a family of subsets of \{1,2,\dots, n\}, and suppose that the symmetric difference between every two sets in \cal F has at most t elements. Then $\cal F$ 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.

Cuts and complete bipartite graphs

The construction of Jeff Kahn and myself can be described as follows:

Construction 1: The ground set is the set of edges of the complete graph on 4p vertices. The family \cal F consists of all subsets of edges which represent the edge sets of a complete bipartite graph with 2p vertices in every part. In this case, n={{4p} \choose {2}}, k=4p^2, and r=2p^2.

It turns out (as observed by A. Nilli) that there is no need to restrict ourselves to  balanced bipartite graphs. A very similar construction which performs even slightly better is:

Construction 2: The ground set is the set of edges of the complete graph on 4p vertices. The family \cal F consists of all subsets of edges which represent the edge set of a complete bipartite graph.

Let f(d) be the smallest integer such that every set of diameter 1 in R^d can be covered by f(d) sets of smaller diameter. Constructions 1 and 2 show that f(d) >exp (K\sqrt d). We would like to replace d^{1/2} by a larger exponent.

The proposed constructions.

To get better bounds for Borsuk’s problem we propose to replace complete bipartite graphs with higher odd-dimensional cocycles.

Construction A: Consider all (2k-1)-dimensional cocycles  of maximum size (or of another fixed size) on the ground set \{1,2,\dots,n\}.

Construction B: Consider all (2k-1)-dimensional cocycles on the ground set \{1,2,\dots,n\}.

A Frankl-Wilson/Frankl-Rodl type problem for cocycles

Conjecture: Let \alpha be a positive real number. There is \beta = \beta (k,\alpha)<1 with the following property. Suppose that

(*) The number of k-cocycles on n vertices with m edges is not zero

and that

(**) m>\alpha\cdot {{n}\choose {k+1}}, and m<(1-\alpha){{n}\choose {k+1}}. (The second inequality is not needed for odd-dimensional cocycles.)

Let \cal F be a family of k-cocycles such that no symmetric difference between two cocycles in \cal F has precisely m sets. Then

|{\cal F}| \le 2^{\beta {{n}\choose {k}}}.

If true even for 3-dimensional cocycles this conjecture will improve the asymptotic lower bounds for Borsuk’s problem.  For example,  if true for 3-cocycles it will imply that f(d) \ge exp (K d^{3/4}). The Frankl-Wilson and Frankl-Rodl theorems have a large number of other applications, and an extension to cocycles may also have other applications.

Crossing number of complete graphs, Turan’s (2k+1,2k) problems, and cocycles

The question on the maximum number of sets in a k-cocycle when k is odd is related to several other (notorious) open problems.

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