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...