Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets

Abstract

For every fixed graph H and every fixed 0 < α < 1, we show that if a graph G has the property that all subsets of size αn contain the “correct” number of copies of H one would expect to find in the random graph G(n,p) then G behaves like the random graph G(n,p); that is, it is p-quasi-random in the sense of Chung, Graham, and Wilson [4]. This solves a conjecture raised by Shapira [8] and solves in a strong sense an open problem of Simonovits and Sós [9].

This is a preview of subscription content, access via your institution.

References

  1. [1]

    F. R. K. Chung and R. L. Graham: Quasi-random set systems, Journal of the AMS4 (1991), 151–196.

    Google Scholar 

  2. [2]

    F. R. K. Chung and R. L. Graham: Quasi-random tournaments, Journal of Graph Theory15 (1991), 173–198.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    F. R. K. Chung and R. L. Graham: Quasi-random hypergraphs, Random Structures and Algorithms1 (1990), 105–124.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    F. R. K. Chung, R. L. Graham and R. M. Wilson: Quasi-random graphs, Combinatorica9(4) (1989), 345–362.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    T. Gowers: Quasirandom groups, Combinatorics, Probability and Computing17 (2008), 363–387.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    M. Krivelevich and B. Sudakov: Pseudo-random graphs, in: More sets, graphs and numbers (E. Györi, G. O. H. Katona and L. Lovász, eds.), Bolyai Society Mathematical Studies Vol. 15 (2006), 199–262.

  7. [7]

    L. Lovász and V. T. Sós: Generalized quasirandom graphs, Journal of Combinatorial Theory Series B98 (2008), 146–163.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    A. Shapira: Quasi-randomness and the distribution of copies of a fixed graph, Combinatorica28(6) (2008), 735–745.

    Article  MathSciNet  Google Scholar 

  9. [9]

    M. Simonovits and V. T. Sós: Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs; Combinatorica17(4) (1997), 577–596.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    A. Thomason: Pseudo-random graphs, in: Proc. of Random Graphs, Poznań 1985, M. Karoński, ed., Annals of Discrete Math. 33 (North Holland 1987), 307–331.

  11. [11]

    A. Thomason: Random graphs, strongly regular graphs and pseudo-random graphs, in: Surveys in Combinatorics (C. Whitehead, ed.), LMS Lecture Note Series 123 (1987), 173–195.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Raphael Yuster.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yuster, R. Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets. Combinatorica 30, 239–246 (2010). https://doi.org/10.1007/s00493-010-2496-0

Download citation

Mathematics Subject Classification (2000)

  • 05C80