The size-Ramsey number of short subdivisions

The r-size-Ramsey number (Formula presented.) of a graph H is the smallest number of edges a graph G can have such that for every edge-coloring of G with r colors there exists a monochromatic copy of H in G. For a graph H, we denote by H^q the graph obtained from H by subdividing its edges with q − 1 vertices each. In a recent paper of Kohayakawa, Retter and Rödl, it is shown that for all constant integers q, r ≥ 2 and every graph H on n vertices and of bounded maximum degree, the r-size-Ramsey number of H^q is at most (Formula presented.), for n large enough. We improve upon this result using a significantly shorter argument by showing that (Formula presented.) for any such graph H.