The multicolour size Ramsey number of a path

Abstract

In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=\Theta((r^2 \log r) \, k).\] Perhaps surprisingly, we do this by improving the lower bound on $\widehat{R}_r(P_k)$.