Upper bounds for multicolour Ramsey numbers

TL;DR

This work establishes a new exponential upper bound for multicolour Ramsey numbers: for every fixed $r \ge 2$ there exists $\delta = \delta(r) > 0$ such that $R_r(k) \le e^{-\delta k} r^{rk}$ for all sufficiently large $k$. The authors introduce a geometric lemma about negative correlation among $r$ vector-valued maps and leverage it through a multicolour book-building framework to create $(t,m)$-books with $t = \varepsilon k$ and $m \approx n / r^t$, enabling an improved ES bound. The approach yields the first exponential improvement over Erdős–Szekeres for $r \ge 3$ and provides a shorter proof for the $r=2$ case in the CGMS line. The results combine high-dimensional geometry with a density-boosting, book-structured method, advancing understanding of multicolour Ramsey numbers and offering new tools for diagonal and off-diagonal bounds alike.

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

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.2
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 3.4
• proof : Proof of Lemma \ref{['lem:lambda']}