On the Ramsey multiplicity of complete graphs

Abstract

We show that, for $n$ large, there must exist at least \[\frac{n^t}{C^{(1+o(1))t^2}}\] monochromatic $K_t$s in any two-colouring of the edges of $K_n$, where $C \approx 2.18$ is an explicitly defined constant. The old lower bound, due to Erd\H{o}s \cite{E62}, and based upon the standard bounds for Ramsey's theorem, is \[\frac{n^t}{4^{(1+o(1))t^2}}.\]