Diagonal Ramsey via effective quasirandomness

We improve the upper bound for diagonal Ramsey numbers to $$R(k+1,k+1)\le \exp (-c(\log k{)}^2)\left(\genfrac{}{}{0px}{}{2k}{k}\right)$$ 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.