Sharper Ramsey lower bounds from refined Gaussian estimates

Abstract

Recently, Ma, Shen and Xie broke the Erdős barrier for off-diagonal Ramsey numbers $R(\ell,C\ell)$, achieving the first exponential improvement over the classical lower bound for every $C>1$ and sufficiently large $\ell$. Hunter, Milojević, and Sudakov later gave a simplified proof using Gaussian random graphs and obtained better quantitative bounds. In this paper we prove a further improvement, and show that the exponent in the Ramsey lower bound can be increased by a strictly positive amount for every fixed $C>1$; as $C\to\infty$, the gain is asymptotically $Θ(p_C^{-1/2}/\log C)$. The improvement is achieved by replacing the subgaussian estimate for truncated Gaussians with a sharp cumulant generating function bound.