Two remarks on the Burr-Erdős conjecture

  • Authors:

  • Jacob Fox;Benny Sudakov

  • Affiliations:

  • Department of Mathematics, Princeton University, Princeton, NJ 08540, United States;Department of Mathematics, UCLA, Los Angeles, CA 90095, United States

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2009

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Abstract

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 Erdos in 1975 conjectured that for each positive integer d there is a constant c"d such that r(H)@?c"dn for every d-degenerate graph H on n vertices. We show that for such graphs r(H)@?2^c^"^d^l^o^g^nn, 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.