On graphs with small ramsey numbers

Let R(G) denote the minimum integer N such that for every bicoloring of the edges of K_N, at least one of the monochromatic subgraphs contains G as a subgraph. We show that for every positive integer d and each γ,0 < γ <1, there exists k = k(d,γ) such that for every bipartite graph G=(W,U,E) with the maximum degree of vertices in W at most d and |U| ≤ |W|^γ, R(G)≤k|W|. This answers a question of Trotter. We give also a weaker bound on the Ramsey numbers of graphs whose set of vertices of degree at least d+1 is independent.