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arXiv · 2410.17197

Upper bounds for multicolour Ramsey numbers

Paul Balister·Béla Bollobás·Marcelo Campos·Simon Griffiths·Eoin Hurley·Robert Morris·Julian Sahasrabudhe·Marius Tiba
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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.

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Paul Balister, Béla Bollobás, Marcelo Campos, Simon Griffiths, Eoin Hurley, Robert Morris, Julian Sahasrabudhe, Marius Tiba. 2024-10-22. Upper bounds for multicolour Ramsey numbers. https://arxiv.org/abs/2410.17197

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