The size-Ramsey number of powers of paths

Given graphs $G$ and $H$ and a positive integer $q$ say that $G$ is $q$-Ramsey for $H$, denoted $Grightarrow (H)_q$, if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The size-Ramsey number $hat{r}(H)$ of a graph $H$ is defined to be $hat{r}(H)=min{|E(G)|colon Grightarrow (H)_2}$. Answering a question of Conlon, we prove that, for every fixed $k$, we have $hat{r}(P_n^k)=O(n)$, where $P_n^k$ is the $k$-th power of the $n$-vertex path $P_n$ (i.e. , the graph with vertex set $V(P_n)$ and all edges ${u,v}$ such that the distance between $u$ and $v$ in $P_n$ is at most $k$). Our proof is probabilistic, but can also be made constructive.