The Combinatorics of Cocycles and Borsuk’s Problem.

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.

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.

A Small Debt Regarding Turan’s Problem

Turan’s problem asks for the minimum number of triangles on n vertices so that every 4 vertices span a triangle. (Or equivalently, for the maximum number of triangles on n vertices without a “tetrahedron”, namely without having four triangles on four vertices.)  When we discussed Turan’s problem, we stated a lemma without giving the proof. Here is the proof.

Lemma: A three uniform hypergraph G on n vertices, so that every four vertices span a triangle satisfies:

$\dim H_1(K,F) \le n-2$.

Here, K is the simplicial complex on the n vertices which has all possible edges and, in addition, the triangles in G. F can be any field of coefficients, below we assume that F=Z/2. (But very little changes are required for the general case.)

Proof:  A little background: The space of 1-dimensional cycles of the complete graph with n vertices is of dimension ${n-1} \choose {2}$. Start with the star T where vertex ‘1’ is adjacent to all other vertices. every edge e={i,j} not in T, determines a cycle c(e) supported by the triangle {1,i} {1,j} and {i,j}. It is easy to see that all these cycles are linearly independent in the space of cycles of the complete graph and span this space.

Now to the proof itself. Let G be a hypergraph with the property that every 4 vertices span a triangle and let K be the 2-dimensional simplicial complex obtained by adding the triangles in G to the complete graph. $H_1(K)$ is still spanned by the cycles c(e) for all edges e={i,j} $1. Let H be a maximal set of edges e so that all the corresponding c(e) are linearly independent in $H_1(K)$. It suffices to prove:

Claim: H is a forest.

Local Events, Turan’s Problem and Limits of Graphs and Hypergraphs

I will write a little about how hectic things are now here at HU, and make two (somewhat related) follow-ups on previous posts: Tell you about Turan’s problem, and about Balázs Szegedi’s lecture from Marburg dealing with limits of graphs and hypergraphs.

Local Events

The second semester at HU started on Sunday, May 11th and it will run until August. This is due to the 3-months Israeli Professors’ strike at the beginning of the academic year. Issues regarding the strike and Israeli academics are quite interesting and we may come back to them. Let me make just one little remark: There is an initiative to transform Israeli universities to a more “market-based” structure. US universities and the new evaluation system in the UK are mentioned as examples, and the Australian academic reforms are often regarded as an act to follow. I was always quite negative about this initiative and skeptical even about the Australian example, and the following post by Terry Tao is telling regarding the Australian reforms. (See also the new blog mathematics in Australia.)

Thia semester I am teaching the basic course in combinatorics and a seminar in probabilistic combinatorics. Continue reading

Extremal Combinatorics I: Extremal Problems on Set Systems

The “basic notion seminar” is an initiative of David Kazhdan who joined HU math department  around 2000. People give series of lectures about basic mathematics (or not so basic at times). Usually, speakers do not talk about their own research and not even always about their field. I gave two lecture series, one about “computational complexity theory” a couple of years ago, and one about extremal combinatorics or Erdös-type combinatorics a few months ago, which later I expanded to a series of five+one talks at Yale. One talk was on  the Borsuk Conjecture,  which I will discuss separately, and five were titled: “Extremal Combinatorics: A working tool in mathematics and computer science.”  Let me try blogging about it. The first talk was devoted to extremal problems concerning systems of sets.

Paul Erdös

1. Three warm up problems

Here is how we move very quickly from very easy problems to very hard problems with a similar flavour.

Problem I: Let  N = {1,2, … , n } . What is the largest size of a family $\cal F$  of subsets of $N$ such that every two sets in $\cal F$ have non empty intersection? (Such a family is called intersecting.)