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Combinatorica

, Volume 17, Issue 2, pp 173–213 | Cite as

OnK4-free subgraphs of random graphs

  • Y. Kohayakawa
  • T. Łuczak
  • V. Rödl
Article

Abstract

For 0<γ≤1 and graphsG andH, writeG→γH if any γ-proportion of the edges ofG spans at least one copy ofH inG. As customary, writeK r for the complete graph onr vertices. We show that for every fixed real η>0 there exists a constantC=C(η) such that almost every random graphG n,p withp=p(n)Cn−2/5 satisfiesG n,p 2/3+ηK4. The proof makes use of a variant of Szemerédi's regularity lemma for sparse graphs and is based on a certain superexponential estimate for the number of pseudo-random tripartite graphs whose triangles are not too well distributed. Related results and a general conjecture concerningH-free subgraphs of random graphs in the spirit of the Erdős-Stone theorem are discussed.

Mathematics Subject Classification (1991)

05C80 05C35 05D99 

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Copyright information

© János Bolyai Mathematical Society 1997

Authors and Affiliations

  • Y. Kohayakawa
    • 1
  • T. Łuczak
    • 2
  • V. Rödl
    • 3
  1. 1.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil
  2. 2.Department of Discrete MathematicsAdam Mickiewicz UniversityPoznańPoland
  3. 3.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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