University of Cambridge > > Combinatorics Seminar > Improved lower bounds for Szemeredi’s theorem.

Improved lower bounds for Szemeredi’s theorem.

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  • UserZach Hunter (ETH Zurich)
  • ClockThursday 19 October 2023, 14:30-15:30
  • HouseMR12.

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Let $r_k(N)$ denote cardinality of the largest subset of $\{1,…,N\}$ which does not contain an arithmetic progression of length $k$. Since 1961, the best lower bound (up to lower order terms) has been due to Rankin, establishing $r_k(N) \ge N exp(-(c_k+o(1)) \log^{p_k}(N))$ for certain explicit constants $c_k,p_k> 0$, generalizing a construction of Behrend.

We shall establish new bounds for this problem, improving the constant $c_k$ for all $k\ge 7$ (with the same value of $p_k$). Our methods also have implications for related problems in finite fields.

This talk is part of the Combinatorics Seminar series.

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