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

In this paper we determine the Ramsey number $R(3,3,k)$ up to a constant factor, showing that $$R(3,3,k)=\mathrm{\Theta }\left(\frac{k^3}{(\log k{)}^2}\right)\mathrm{.}$$ 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 $R_r(3,\dots ,3,k)$ up to a constant factor for every fixed $r⩾3$.