# 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 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) (1 + c), for some constant c > 0.

This talk will describe a new work showing that the same holds when C ≈ 2(log N)0.09, thus getting closer to Behrend's construction.

Based on joint work with Zander Kelley.

### Date & Time

March 20, 2023 | 10:00am – 11:00am

### Location

Simonyi 101 and Remote Access

Zander Kelley

### Speaker Affiliation

University of California, Los Angeles; University of Illinois Urbana-Champaign