Ramsey numbers of degenerate graphs

A graph is d-degenerate if all its subgraphs have a vertex of degree at most d. We prove that there exists a constant c such that for all natural numbers d and r, every d-degenerate graph H of chromatic number r with |V(H)| >= 2(d22cr) has Ramsey number at most 2(d22cr) |V(H)|. This solves a conjecture of Burr and Erdos from 1973.