University of Cambridge > Talks.cam > Discrete Analysis Seminar > Sets without four term progressions but rich in three term progressions

Sets without four term progressions but rich in three term progressions

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If you have a question about this talk, please contact Aled Walker.

The main question that will be addressed in this talk is the following: given a set A which does not contain any four term arithmetic progressions, is it necessarily the case that there exists a large subset of A which does not contain any three term arithmetic progressions?

Perhaps one might guess that the answer is “yes”, and that by deleting a relatively small number of elements from A we can destroy all progressions. In fact this rough intuition seems to be false, as we aim to show in this talk by constructing sets (in both the integers and finite field setting) with no 4APs but for which all large subsets contain a 3AP. Possible connections with quantitative bounds for Roth’s Theorem will also be discussed. The proof uses the method of hypergraph containers.

This talk is based on joint work with Cosmin Pohoata

This talk is part of the Discrete Analysis Seminar series.

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