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

TL;DR

This paper surveys the history of bounds on diagonal Ramsey numbers, highlighting recent improvements that reduced the upper bound from exponential to approximately 3.993^k, and discusses new proof techniques.

Contribution

It presents the recent breakthrough establishing a tighter upper bound on diagonal Ramsey numbers and compares different proof methods used in this advancement.

Findings

Proved that N=3.993^k suffices for diagonal Ramsey numbers.

Reviewed historical bounds from factorial to exponential.

Discussed alternative, more conceptual proofs of the new bound.

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.