The size‐Ramsey number of short subdivisions

The r ‐size‐Ramsey numberR ^r ( H ) 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( log n ) 20 ( q − 1 )n 1 + 1 / q, for n large enough. We improve upon this result using a significantly shorter argument by showing thatR ^r ( H q ) ≤ O ( n 1 + 1 / q ) for any such graph H .