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)=Θ((r^2 \log r) \, k).] Perhaps surprisingly, we do this by improving the lower bound on $\widehat{R}_r(P_k)$.