  Given graphs $G$ and $H$ and a positive integer $q$ say that $G$ is
$q$-Ramsey for $H$, denoted $G\rightarrow (H)_q$, 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 $\hat{r}(H)=\min\{|E(G)|\colon
G\rightarrow (H)_2\}$. 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.
