Upper bounds on diagonal Ramsey numbers [after Campos, Griffiths, Morris, and Sahasrabudhe]

Abstract

Ramsey's theorem states that if $N$ is sufficiently large, then no matter how one colors the edges among $N$ vertices with two colors, there are always $k$ vertices spanning edges in only one color. Given this theorem, it is natural to ask ``how large is sufficiently large?'' Ramsey's original proof showed that $N=k!$ is sufficient, and five years later Erd\H{o}s and Szekeres improved this bound to $N=4^k$. And then progress stalled for almost 90 years. In this survey, I present the history of the problem and discuss some of the ideas used in the recent breakthrough of Campos--Griffiths--Morris--Sahasrabudhe, who proved that $N=3.993^k$ is sufficient. In addition, I discuss the subsequent work of Balister, Bollob\'as, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba, who gave an alternative, and more conceptual, proof.