Two remarks on the Burr–Erdős conjecture

The Ramsey number r ( H ) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K N 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 c d such that r ( H ) ≤ c d n for every d -degenerate graph H on n vertices. We show that for such graphs r ( H ) ≤ 2 c d log n 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.