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

OnK4-free subgraphs of random graphs

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


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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  1. [1]
    L. Babai, M. Simonovits, J. H. Spencer: Extremal subgraphs of random graphs,J. Graph Theory,14 (1990), 599–622.Google Scholar
  2. [2]
    B. Bollobás:Extremal Graph Theory, Academic Press, London, 1978.Google Scholar
  3. [3]
    B. Bollobás:Random Graphs, Academic Press, London, 1985.Google Scholar
  4. [4]
    F. R. K. Chung: Subgraphs of a hypercube containing no small even cycles,J. Graphs Theory,16 (1992), 273–286.Google Scholar
  5. [5]
    P. Frankl, V. Rödl: Large triangle-free subgraphs in graphs withoutK 4,Graphs and Combinatorics,2 (1986), 135–244.Google Scholar
  6. [6]
    Z. Füredi: Random Ramsey graphs for the four-cycle,Discrete Maths.,126 (1994), 407–410.Google Scholar
  7. [7]
    P. E. Haxell, Y. Kohayakawa, T. Łuczak: The induced size-Ramsey number of cycles,Combinatorics, Probability, and Computing,4 (1995), 217–239.Google Scholar
  8. [8]
    P. E. Haxell, Y. Kohayakawa, T. Łuczak: Turán's extremal problem in random graphs: forbidding odd cycles,Combinatorica,16 (1996), 107–122.Google Scholar
  9. [9]
    P. E. Haxell, Y. Kohayakawa, T. Łuczak: Turán's extremal problem in random graphs: forbidding even cycles.J. Combin. Theory, Ser. B.,64 (1995), 273–287.Google Scholar
  10. [10]
    S. Janson: Poisson approximation for large deviations,Random Structures and Algorithms,1 (1990), 221–230.Google Scholar
  11. [11]
    Y. Kohayakawa: Szemerédi's regularity lemma for sparse graphs, in:Foundations of Computational Mathematics (eds.: F. Cucker, M. Shub), 1997, Berlin, Heidelberg, Springer-Verlag, 216–230.Google Scholar
  12. [12]
    Y. Kohayakawa, B. Kreuter, A. Steger: An extremal problem for random graphs and the number of graphs with large even-girth, (1995) submitted.Google Scholar
  13. [13]
    Y. Kohayakawa, T. Łuczak, V. Rödl: Arithmetic progressions of length three in subsets of a random set,Acta Arithmetica,LXXV (1996), 133–163.Google Scholar
  14. [14]
    P. Rödl, A. Ruciński: Lower bounds on probability thresholds for Ramsey properties, in:Combinatorics—Paul Erdős is Eighty (Volume 1) (eds.: D. Miklós, V. T. Sós, T. Szőnyi), Budapest, Bolyai Soc. Math. Studies, 1993, 317–346.Google Scholar
  15. [15]
    V. Rödl, A. Ruciński: Threshold functions for Ramsey properties,J. Amer. Math Soc.,8 (1995), 917–942.Google Scholar
  16. [16]
    E. Szemerédi: Regular partitions of graphs, in:Problèmes Combinatoires et Théorie des Graphes, Proc. Colloque Inter. CNRS (eds.: J.-C. Bermond, J.-C., Fournier, M. Las Vergnas, D. Sotteau), CNRS, Paris, 1978, 399–401.Google Scholar

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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