Optimizing the CGMS upper bound on Ramsey numbers

TL;DR

This paper simplifies and extends recent advances in bounding Ramsey numbers, providing a shorter proof and improved bounds, including a new upper bound for diagonal Ramsey numbers that extends to multicolor cases.

Contribution

It introduces a simplified inductive proof method that improves upper bounds on Ramsey numbers and clarifies parameter dependencies, extending results to multicolor scenarios.

Findings

Shorter proof of upper bounds on Ramsey numbers

Extended bounds to multicolor Ramsey numbers

Established an explicit upper bound R(k,k) ≤ (3.8)^{k+o(k)}

Abstract

In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the diagonal Ramsey numbers since 1935. We shorten their proof, replacing the underlying book algorithm with a simple inductive statement. This modification allows us - to give a very short proof of an improved upper bound on the off-diagonal Ramsey numbers, which extends to the multicolor setting, and - to clarify the dependence of the bounds on underlying parameters and optimize these parameters, obtaining, in particular, an upper bound $$R(k,k) \leq (3.8)^{k+o(k)}$$ on the diagonal Ramsey numbers.