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 $r\ge 2$ and $k\ge 100\log r$, we have \[ \widehat{R}_r(P_k)=Θ((r^2 \log r) \, k).\] Perhaps surprisingly, we do this by improving the lower bound on ${\hat{R}}_r(P_k)$.

11+3 pages. Exposition streamlined