Note on the multicolour size-Ramsey number for paths

The size-Ramsey number $\widehat{R}(F,r)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\widehat{R}(P_n,r)=O((\log r)r^{2}n)$. This upper bound is nearly optimal, since it is also known that $\widehat{R}(P_n,r)=\mathrm{\Omega }(r^{2}n)$.