DAN

Abstract

In 1996 Kohayakawa, Luczak and Rödl proved that Roth's theorem holds almost surely inside a subset of {1,2,...,n} of density Cn^{-1/2}. That is, if A is such a subset, chosen randomly, then with high probability every subset B of A of size at least c|A| contains an arithmetic progression of length 3. (The constant C depends on c.) It is easy to see that the result fails for sparser sets A. Recently, David Conlon and I found a new proof of this theorem using a very general method. As a consequence we obtained many other results with sharp bounds, thereby solving several open problems. In this talk I shall focus on the case of Roth's theorem, but the generality of the method should be clear from that.