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 Erdos, Frankl and Rödl (Graphs Combin. 2 (1986) 113), we prove that A figure is presented. 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 Szemerédi's Regularity Lemma and the stability theorem of Erdos and Simonovits. The bound above is essentially best possible. © 2003 Elsevier Inc. All rights reserved.
Journal of Combinatorial Theory. Series B
Balogh, J., Bollobás, B., & Simonovits, M. (2004). The number of graphs without forbidden subgraphs. Journal of Combinatorial Theory. Series B, 91 (1), 1-24. https://doi.org/10.1016/j.jctb.2003.08.001