On the Ramsey multiplicity of complete graphs

TL;DR

This paper establishes a new lower bound on the number of monochromatic complete subgraphs in two-colored complete graphs, improving upon previous bounds by explicitly defining a smaller constant in the asymptotic estimate.

Contribution

The authors derive an improved asymptotic lower bound for the Ramsey multiplicity of complete graphs, with an explicitly calculated constant less than 4.

Findings

New lower bound with constant C ≈ 2.18

Improved asymptotic estimate over Erdős's bound

Explicit constant enhances understanding of monochromatic subgraph frequency

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 \[\frac{n^t}{4^{(1+o(1))t^2}}.\]