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

Abstract

In this paper we determine the Ramsey number $R(3,3,k)$ up to a constant factor, showing that $$R(3,3,k) = \Theta\bigg( \frac{k^3}{(\log k)^2} \bigg).$$ The proof of the lower bound combines the Hefty-Horn-King-Pfender construction for $R(3,k)$ with the method of Alon and R\"odl. 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 $r \geqslant 3$.