The number of graphs without forbidden subgraphs

  • Authors:

  • József Balogh;Béla Bollobás;Miklós Simonovits

  • Affiliations:

  • Mathematical Institute, Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127, Hungary;Mathematical Institute, Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127, Hungary;Mathematical Institute, Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127, Hungary

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2004

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Abstract

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 Erdös, Frankl and Rödl (Graphs Combin. 2 (1986) 113), we prove that |P(n, L)|≤ 2½(1-1/p)n2 . O(n2-σ) 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