The multicolour size-Ramsey number of powers of paths

Given a positive integer s, a graph G 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 ȓs(H) of a graph H is defined to be ȓs(H)= min{|E(G)|: G→(H)s}. We prove that, for all positive integers k and s, we have ȓs(Pnk)=O(n), where Pnk is the kth power of the n-vertex path Pn.