The number of graphs without forbidden subgraphs

Given a family L of graphs, set p=p(L)=minL∈L χ(L)−1 and, for n⩾1, denote by P(n,L) the set of graphs with vertex set [n] containing no member of L as a subgraph, and write ex(n,L) for the maximal size of a member of P(n,L). Extending a result of Erdős, Frankl and Rödl (Graphs Combin. 2 (1986) 113), we prove that|P(n,L)|⩽2121−1pn2+O(n2−γ)for some constant γ=γ(L)>0, and characterize γ in terms of some related extremal graph problems. In fact, if ex(n,L)=O(n2−δ), then any γ<δ will do. Our proof is based on Szemerédiʹs Regularity Lemma and the stability theorem of Erdős and Simonovits. The bound above is essentially best possible.