# Joint TCS - Combinatorics Seminar

Wednesday, March 22, 2023 at 3:00pm to 5:00pm

MIT-Math Dept., Room 2-449

**Speaker**: Raghu Meka (UCLA)

**Title**: Strong bounds for 3-progressions

**Abstract**:

Suppose you have a set S of integers from \{1,2,...,*N* \ that contains at least* N / C* elements. Then for large enough *N* , must *S* contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?

In 1953, Roth showed that this is indeed the case when *C* > \Omega(\log \log *N*), while Behrend in 1946 showed that C can be at most 2^{O(\sqrt{\log N})}. Since then, the problem has been a cornerstone of the area of additive combinatorics. Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on* C* to *C* = (\log *N*)^{1+c}, for some constant *c* > 0.

This talk will describe a new work that* C* >2^{\Omega((\log N)^{0.09), thus getting closer to Behrend's construction. Based on joint work with Zander Kelley.

The first hour of the talk will be self-contained and describe the main ideas of the proof. The second hour will be a deeper follow-up of some elements of the proof.

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