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arXiv · 2609.28455

On the multicolour Ramsey numbers $R(3,3,k)$

Bruno Andrades·Marcelo Campos·Robert Morris
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Abstract

In this paper we determine the Ramsey number $R(3,3,k)$ up to a constant factor, showing that $$R(3,3,k) = Θ\bigg( \frac{k^3}{(\log k)^2} \bigg).$$ The proof of the lower bound combines the Hefty-Horn-King-Pfender construction for $R(3,k)$ with the method of Alon and Rödl. Using the same proof, we also determine the $r$-colour Ramsey numbers $R_r(3,\ldots,3,k)$ up to a constant factor for every fixed $r \geqslant 3$.

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Bruno Andrades, Marcelo Campos, Robert Morris. 2026-09-23. On the multicolour Ramsey numbers $R(3,3,k)$. https://arxiv.org/abs/2609.28455

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