Around Borsuk’s Conjecture 1: Some Problems

Greetings to all!

Karol Borsuk conjectured in 1933 that every bounded set in R^d can be covered by d+1 sets of smaller diameter. In a previous post I described the counterexample found by Jeff Kahn and me. I will devote a few posts to related questions that are still open. I will list and discuss such questions in the first and second  parts. In the third part I will describe an approach towards better examples which is related to interesting extremal combinatorics. (Of cocyclec. this post appeared already.) In the fourth part I will try to “save the conjecture”, namely to present a variation of the conjecture which might be true. Let f(d) be the smallest integer so that every set of diameter one in R^d can be covered by f(d) sets of smaller diamete

Continue reading

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.

Continue reading

Tentative Plans and Belated Updates II


Elementary school reunion: Usually, I don’t write about personal matters over the blog, but having (a few weeks ago) an elementary school reunion after 42 years was a moving and exciting event as to consider making an exception. For now, here is a picture:


Jirka’s Miraculous year

It looks like a lot is happening. From time to time I think that I should tell on my blog about exciting new things I hear about, but this is quite a difficult task. Perhaps I should at least post updates about progress on problems I discussed earlier, but even this is not easy.  Jirka Matousek wrote a paper entitled The dawn of an algebraic era in discrete geometry?  The paper starts as follows:

To me, 2010 looks as annus mirabilis, a miraculous year, in several areas of my mathematical interests. Below I list seven highlights and breakthroughs, mostly in discrete geometry, hoping to share some of my wonder and pleasure with the readers.

The paper lists seven startling new results. A few of these results were discussed here, a few others I have planned to discuss later, and yet a few others (like the recent solution by June Huh of the famous unimodularity conjecture for the coefficients of chromatic polynomials of graphs) caught me by a complete surprise. (Here is a link to a follow up paper by June Huh and Eric Katz.) Let me add one additional item, namely the solution (in the negative) by Boris Bukh of Eckhoff’s partition conjecture.

Other wonderful combinatorics news

These are also good times for other areas of combinatorics. I described some startling developments (e.g., here and here and here) and there is more. There were a few posts (here and here) on the Cup Set Problem. Recently Michael Bateman and Nets Katz improved, after many years, the Roth-Meshulam bound.  See these two posts on Gowers’s blog (I,II). Very general theorems discovered independently by David Conlon and Tim Gowers and by Matheas Schacht show that many theorems (such as Ramsey’s theorem or Turan’s theorem) continue to hold for substructures of sparse random sets.  Louis Esperet, Frantisek Kardos, Andrew King, Daniel Kral, and Serguei Norine proved the Lovasz-Plummer conjecture. They showed  that every cubic bridgeless graph G has at least 2^(|V(G)|/3656) perfect matchings. The concept of flag algebras, discovered by Razborov, is an extremely useful tool for extremal set theory. It has led to solutions of several problems and seems to bring us  close to a solution of  Turan’s Conjecture (which  we discussed here and here.) For example, it led to the solution by Hamed Hatami, Jan Hladký, Daniel Král, Serguei Norine, and Alexander Razborov of the question on the maximum number of pentagons in triangle-free graphs.  Hamed Hatami found a structure theorem for Boolean functions with coarse thresholds w.r.t. small probabilities. This extends and sharpens results by Ehud Friedgut and Jean Bourgain. I finally caught up (thanks to Reshef Meir) with Moser-Tardos result giving a new algorithmic proof for Lovasz local lemma. Amazing! You can read about it here.

Some updates on my Internet questions

Imre Leader and Eoin Long wrote a paper entitled tilted Sperner families, which solves a question I raised in the context of polymath1. Imre and Eoin give additional results and conjectures. My motivation was to try to come up (eventually) with very very general conjectures which include density Hales-Jewett as a very special case and are also related to error-correcting codes. Raman Sanyal discovered a dual form of Tverberg’s theorem in terms of families of fans.  (We asked about it here.) There is a new paper on the Entropy-Influence conjecture entitled The Fourier Entropy-Infuence Conjecture for certain classes of Boolean functions, by Ryan O’Donnell, John Wright, and Yuan Zhou, The paper contains a proof of the conjecture for symmetric Boolean functions and  various other cases. This is the first new result on the conjecture for many years. Also there is nice progress for the AC^0-prime number conjecture asked about in a previous post, and a subsequent Mathoverflow question (There I will keep updating matters.). Ben Green solved the conjecture! Jean Bourgain settled the more general MO question and also found results on certain AC(2) circuits.

Newton Institute and Oberwolfach,

And it seems that things are moving along nicely in other areas close to my heart.  A week ago (This was actually two months ago)  I participated in a workshop at the Newton Institute on discrete harmonic analysis. And in the first week of February we had our traditional Oberwolfach meeting on geometric and topological combinatorics. Many interesting results!

A visit to IQI

In the last week of January I visited Caltech. I missed the IPAM meeting scheduled a week before because my visa arrived too late but I still made it to IQI. This was a very nice opportunity as most of my time at Caltech was devoted to quantum information/quantum computation issues related to my own work on quantum fault tolerance. So I gave an “informal” seminar describing my point of view (and gave it again the next day at USC). Here are the slides.  My lecture was followed by two-hour discussion of the more technical details of my conjectures, seeking weak points and counterexamples in what I said, and trying to associate it with physics. There followed further discussions about some aspects of quantum fault tolerance and more general questions of quantum information with John Preskill, Leonard Schulman, Daniel Lidar and a few other people. (A lot of brilliant young people!) I learned quite a lot and was happy with this opportunity. (Of course, I did talk a little with Caltechian old and new friends about bona fide combinatorics questions.)

Three jokes for a dollar

On the weekend I took the LA metro (whose mere existence surprised me) and visited downtown LA and Hollywood. Next to Pershing station a person stopped me and asked if I want to buy three jokes for one dollar. I first said no, but then I reconsidered, called him back and gave him a dollar. To his disappointment and mine I did not understand the first joke, but the other two (perhaps adjusted to my revealed level of understanding) were quite good.

Hectic semester at HUJI

Here at Huji things are as hectic as always with 10-15 weekly research seminars. This week (This was two months ago; the semester have just ended). Continue reading

Another way to Revolutionize Football

The angle of Victoria Beckham’s hat (here in a picture from a recent wedding) is closely related to our previous post on football

One of the highlights of the recent Newton Institute  conference on discrete harmonic analysis was a football game which was organized by Frank Barthe and initiated independently by Barthe and Prasad Tetali. There were two teams of 10 players (more or less), I was the oldest player on the field, and it was quite exciting. No spontaneous improvement of my football skills has occurred since my youth.

All the lectures at the conference were videotaped and can be found here. (The football game itself was not videotaped.) Let me mention an idea for a new version of football which occurred to me while playing. For an early suggested football revolution and some subsequent theoretical discussions see this post on football and the intermediate value theorem.

The New Football Game

There are four teams. Team L (the left team), Team R (the right team), Team D (the defense team) and team O (the offense team.)  The left team protects the left goal and tries to score the right goal, the right team protects the right goal and tries to score the left goal, the defense team tries to prevent goals on both sides of the field and the offense team tries to score as much as possible goals on both sides of the field.

In formal terms,  define X to be the number of goals scored to the right and Y the number of goals scored to the left. Then the four teams try to maximize X-Y, Y-X, -X-Y and X+Y, respectively. (I do not assume teams A and B change position at halftime, but the formula can easily be adjusted.)

We can start with five players on each team (but this is negotiable.) Continue reading