A Couple Updates on the Advances-in-Combinatorics Updates

In a recent post I mentioned quite a few remarkable recent developments in combinatorics. Let me mention a couple more.

Independent sets in regular graphs

A challenging conjecture by Noga Alon and Jeff Kahn in graph theory was about the number of independent sets in regular graphs. The conjecture asserts that the number of independent sets in an N-vertex d-regular graph is at most (2^{d+1}-1)^{N/2d}. Alon proved in 1991 a weaker form of the conjecture that was posed by Granville. Kahn proved in 2001 the conjecture for bipartite graphs. In 2010, Yufei Zhao, an undergraduate student, proved the conjecture.  (The number of independent sets in a regular graph, Combinatorics, Probability and Computing 19 (2010), 315-320. A link to the arxived paper.) The proof is based on a surprising reduction to the bipartite case.  Zhao’s result came as a surprise to several experts in the field who were working on this problem.

Around Roth’s theorem

The second development is about Roth’s theorem that we often discuss (E.g., here, and here and here).

Suppose that R_n is a subset of \{1,2,\dots, n \} of maximum cardinality not containing an arithmetic progression of length 3. Let g(n)=n/|R_n|. The gap between the upper and lower bounds for g(n) is exponential. Can we look behind the horizon and make an educated guess about the true behavior of g(n)?

recent paper entitled “Roth’s theorem in many variables” by Tomasz Schoen and  Ilya D. Shkredov contains a remarkable piece of evidence which is strong enough for me to update my beliefs on the problem.

Schoen and Shkredov proved that if a subset A of {1, 2,…, N} has no nontrivial solution to the equation x_1+x_2+x_3+x_4+x_5=5y then the cardinality of A is at most N e^{-c(log N)^{1/7-t}}, where t>0 is an arbitrary number, and c>0 is an absolute constant. In view of the well-known Behrend construction their estimate is close to best possible.

Roth’s theorem is about the equation x_1+x_2=2y. I used to think that much better bounds than Behrend are quite possible. (But I was aware that most experts have the opposite view.) Schoen and Shkredov’s result is a strong piece of evidence that the truth is near the Behrend’s bound. Two comments: First, if there are conceptual differences between the case of 6 variables and the case of 3 variables I will be happy to know about them.   Second, additional interesting examples of sets without 3-term AP are still very welcome.