# Computer Science/Discrete Mathematics Seminar I

## Strong Bounds for 3-Progressions

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 ≈ (log **** log **** N)*, while Behrend in 1946 showed that *C* can be at most *2 ^{Ω(√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)*, for some constant

^{(1 + c)}*c > 0*.

This talk will describe a new work showing that the same holds when *C ≈ 2 ^{(log }*

^{}*, thus getting closer to Behrend's construction.*

^{ N)0.09}Based on joint work with Zander Kelley.