Upper bounds for multicolour Ramsey numbers

Abstract

The $r$-colour Ramsey number $R_r(k)$ is the minimum $n\in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r⩾2$, that $$R_r(k)⩽e^{-\hat{I}´k}{r}^{rk}$$ for some constant $\hat{I}´=\hat{I}´(r)>0$ and all sufficiently large $k\in \mathbb{N}$. For each $r⩾3$, this is the first exponential improvement over the upper bound of Erdős and Szekeres from 1935. In the case $r=2$, it gives a different (and significantly shorter) proof of a recent result of Campos, Griffiths, Morris and Sahasrabudhe.

17 pages, minor revision