Combinatorial theorems in sparse random sets

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• Tim Gowers (Cambridge)
• Thursday 04 December 2008, 14:30-15:30
• MR12.

If you have a question about this talk, please contact Andrew Thomason.

Let us call a set X of integers (delta,k)-Szemer’edi if every subset Y of X that contains at least delta|X| elements contains an arithmetic progression of length k. Suppose that X is a random subset of {1,2,...,n} with each element chosen independently with probability p. For what values of p is there a high probability that X is (delta,k)-Szemer’edi?

There is a trivial lower bound of cn^{-1/(k-1)} (since at this probability there will be many fewer progressions than there are points in the set). We match this to within a constant by a new upper bound. There are many other conjectures and partial results of this kind in the literature: our method is very general and seems to deal with them all. A key tool in the proof is the finite-dimensional Hahn-Banach theorem. This is joint work with David Conlon.

This talk is part of the Combinatorics Seminar series.

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