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Proving theorems inside sparse random sets

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If you have a question about this talk, please contact Mustapha Amrani.

Discrete Analysis

In 1996 Kohayakawa, Luczak and Rdl 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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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