On the multicolor Ramsey number of a graph with mm edges

The multicolor Ramsey number rk(F)rk(F) of a graph FF is the least integer nn such that in every coloring of the edges of KnKn by kk colors there is a monochromatic copy of FF. In this short note we prove an upper bound on rk(F)rk(F) for a graph FF with mm edges and no isolated vertices of the form k6km2/3k6km2/3 addressing a question of Sudakov (2011). Furthermore, the constant in the exponent in the case of bipartite FF and two colors is lowered so that r2(F)≤2(1+o(1))22m improving the result of Alon, Krivelevich and Sudakov (2003).