A note on pseudorandom Ramsey graphs

TL;DR

This paper establishes new bounds on multicolor Ramsey numbers using pseudorandom graphs, improving previous results for cycles and general $K_s$-free graphs, with implications for understanding graph Ramsey theory.

Contribution

It proves that optimal $K_s$-free pseudorandom graphs imply specific asymptotic bounds for Ramsey numbers and improves lower bounds for cycles, introducing novel methods and results.

Findings

Proves $r(s,t) = t^{s-1+o(1)}$ assuming optimal pseudorandom graphs.

Improves lower bounds for $r(C_{ ext{odd } ext{and} ext{ specific } ext{cycles}})$ by polylogarithmic factors.

Provides first known improvements over random process bounds for certain cycle Ramsey numbers.

Abstract

For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs exist, then the Ramsey number $r(s,t) = t^{s-1+o(1)}$ as $t \rightarrow \infty$. Our method also improves the best lower bounds for $r(C_{\ell},t)$ obtained by Bohman and Keevash from the random $C_{\ell}$-free process by polylogarithmic factors for all odd $\ell \geq 5$ and $\ell \in \{6,10\}$. For $\ell = 4$ it matches their lower bound from the $C_4$-free process. We also prove, via a different approach, that $r(C_5, t)> (1+o(1))t^{11/8}$ and $r(C_7, t)> (1+o(1))t^{11/9}$. These improve the exponent of $t$ in the previous best results and appear to be the first examples of graphs $F$ with cycles for which such an improvement of the exponent for $r(F, t)$ is shown over the bounds given by the random $F$-free process and random graphs.