Diagonal Ramsey via effective quasirandomness

Abstract

We improve the upper bound for diagonal Ramsey numbers to

$$R(k+1,k+1)\le exp(-c{(logk)}^2)\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)$$

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.