TY - JOUR

T1 - The number of graphs without forbidden subgraphs

AU - Balogh, József

AU - Bollobás, Béla

AU - Simonovits, Miklós

N1 - Funding Information:
E-mail addresses: jobal@sol.cc.u-szeged.hu (J. Balogh), bollobas@msci.memphis.edu (B. Bollobás), miki@renyi.hu (M. Simonovits). 1Partially supported by NSF Grant DMS-9971788, ITR Grant 0225610, and DARPA Grant F33615-01-C-1900. 2Partially supported by OTKA Grants T026069, T038210.

PY - 2004/5

Y1 - 2004/5

N2 - 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.

AB - 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.

KW - Erdos-Kleitman-Rothschild theory

KW - Extremal graphs

KW - Speed function

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U2 - 10.1016/j.jctb.2003.08.001

DO - 10.1016/j.jctb.2003.08.001

M3 - Article

AN - SCOPUS:2342449349

VL - 91

SP - 1

EP - 24

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -