Upper bounds for multicolour Ramsey numbers

TL;DR

This paper establishes new exponential upper bounds for multicolour Ramsey numbers, improving previous results significantly for three or more colours and providing a shorter proof for the two-colour case.

Contribution

It proves the first exponential improvement over Erdős and Szekeres' bounds for multicolour Ramsey numbers with three or more colours, and offers a new proof for the two-colour case.

Findings

For fixed r ≥ 2, R_r(k) ≤ e^{- ext{delta} k} r^{rk} for large k.

First exponential improvement over 1935 bounds for r ≥ 3.

Shorter proof of recent two-colour Ramsey number bounds.

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 $r \geqslant 2$, that $$R_r(k) \leqslant e^{-\delta k} r^{rk}$$ for some constant $\delta = \delta(r) > 0$ and all sufficiently large $k \in \mathbb{N}$. For each $r \geqslant 3$, this is the first exponential improvement over the upper bound of Erd\H{o}s 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.