Two remarks on the Burr-Erdos conjecture

The Ramsey number r(H) of a graph H is the minimum positive integer N such that every twocoloring of the edges of the complete graph KN on N vertices contains a monochromatic copy of H . A graph H is d-degenerate if every subgraph of H has minimum degree at most d. Burr and Erdős in 1975 conjectured that for each positive integer d there is a constant cd such that r(H) ≤ cdn for every d-degenerate graph H on n vertices. We show that for such graphs r(H) ≤ 2d √ log n, improving on an earlier bound of Kostochka and Sudakov. We also study Ramsey numbers of random graphs, showing that for d fixed, almost surely the random graph G(n, d/n) has Ramsey number linear in n. For random bipartite graphs, our proof gives nearly tight bounds.