OnK 4-free subgraphs of random graphs


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+η K 4. 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.

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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.

  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 

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Additional information

The first author was partially supported by FAPESP (Proc. 93/0603-1) and by CNPq (Proc. 300334/93-1 and ProTeM-CC-II Project ProComb). Part of this work was done while the second author was visiting the University of São Paulo, supported by FAPESP (Proc. 94/4276-8). The third author was partially supported by the NSF grant DMS-9401559.

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Kohayakawa, Y., Łuczak, T. & Rödl, V. OnK 4-free subgraphs of random graphs. Combinatorica 17, 173–213 (1997). https://doi.org/10.1007/BF01200906

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Mathematics Subject Classification (1991)

  • 05C80
  • 05C35
  • 05D99