The number of K m,m -free graphs

Abstract

A graph is called H-free if it contains no copy of H. Denote by f n (H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that f n (H) ≤ 2(1+o(1))ex(n,H). This was first shown to be true for cliques; then, Erdős, Frankl, and Rodl proved it for all graphs H with χ(H)≥3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log2 f n (H) is not known. We prove that f n (K m,m ) ≤ 2O (n 2−1/m ) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m∈{2,3}, and possibly for all other values of m, for which the order of ex(n,K m,m ) is conjectured to be Θ(n 2−1/m ). Our method also yields a bound on the number of K m,m -free graphs with fixed order and size, extending the result of Furedi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Łuczak and show that almost all K 3,3-free graphs of order n have more than 1/20·ex(n,K 3,3) edges.