An update on multicolor Ramsey lower bounds

TL;DR

This paper improves lower bounds on multicolor Ramsey numbers by leveraging recent advances in spherical random geometric graph constructions, resulting in a small exponential enhancement over previous bounds.

Contribution

It introduces a new connection between spherical geometric graphs and Ramsey bounds, leading to improved lower bounds for multicolor Ramsey numbers.

Findings

Derived a new exponential lower bound for multicolor Ramsey numbers.

Connected spherical random geometric graph constructions to Ramsey theory.

Achieved a small but significant improvement over existing bounds.

Abstract

Building upon previous works by Conlon-Ferber and Wigderson, Sawin showed a few years ago that upper bounds on the minimum density of independent sets in a $K_t$-free $G$ can be used to provide lower bounds for multicolor Ramsey numbers. In this note, we observe how a further improved upper bound on this parameter directly follows from a recent spherical random geometric graph construction of Ma-Shen-Xie. As a consequence, we derive a small exponential improvement over the best known lower bounds for multicolor Ramsey numbers.