The multicolour size-Ramsey number of powers of paths

Given a positive integer s, a graph G is s-Ramsey for a graph H, denoted G→(H)s, if every s-colouring of the edges of G contains a monochromatic copy of H. The s-colour size-Ramsey number rˆs(H) of a graph H is defined to be rˆs(H)=min⁡{|E(G)|:G→(H)s}. We prove that, for all positive integers k and s, we have rˆs(Pnk)=O(n), where Pnk is the kth power of the n-vertex path Pn.