Diagonal Ramsey via effective quasirandomness

TL;DR

This paper advances the upper bounds for diagonal Ramsey numbers by developing an optimal effective quasirandomness framework, demonstrating a natural limit to how much these bounds can be improved.

Contribution

It introduces an improved upper bound for diagonal Ramsey numbers using an effective quasirandomness approach, extending previous frameworks and establishing a natural barrier.

Findings

New upper bound: R(k+1,k+1) ≤ exp(-c(log k)^2) * binom(2k,k) for k ≥ 3

Demonstrates optimality of the quasirandomness convergence results

Establishes a natural barrier to further improvements in bounds

Abstract

We improve the upper bound for diagonal Ramsey numbers to \[R(k+1,k+1)\le\exp(-c(\log k)^2)\binom{2k}{k}\] for $k\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal "effective quasirandomness" results about convergence of graphs. This optimality represents a natural barrier to improvement.