Diagonal Ramsey via effective quasirandomness

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.

combinatoricsnumber theory

Audience: researchers in the topic

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