The number of Km, m-free graphs

A graph is called H-free if it contains no copy of H. Denote by fn(H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that fn(H) ≤ 2(1+o(1)) ex(n,H). This was first shown to be true for cliques; then, Erdős, Frankl, and Rödl 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 fn(H) is not known. We prove that fn(Km,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,Km,m) is conjectured to be Θ(n2−1/m). Our method also yields a bound on the number of Km,m-free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Luczak and show that almost all K3,3-free graphs of order n have more than 1/20 · ex(n,K3,3) edges.