Hypergraphs with independent neighborhoods

Abstract

For each k ≥ 2, let ρ k ∈ (0, 1) be the largest number such that there exist k-uniform hypergraphs on n vertices with independent neighborhoods and (ρ k + o(1))( n k ) edges as n → ∞. We prove that ρ k = 1 − 2logk/k + Θ(log log k/k) as k → ∞. This disproves a conjecture of Füredi and the last two authors.

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References

  1. [1]

    B. Bollobás: Three-graphs without two triples whose symmetric difference is contained in a third, Discrete Math. 8 (1974), 21–24.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    B. Bollobás: Random Graphs, 2nd Edition, Cambridge University Press, 2001.

  3. [3]

    F. Chung and L. Lu: An upper bound for the Turán number t3(n, 4), J. Combin. Theory Ser. A87(2) (1999), 381–389.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    D. de Caen: The current status of Turán’s problem on hypergraphs, in: Extremal problems for finite sets (Visegrád, 1991), Bolyai Soc. Math. Stud., vol. 3, János Bolyai Math. Soc., Budapest, 1994, pp. 187–197.

    Google Scholar 

  5. [5]

    P. Frankl and Z. Füredi: A new generalization of the Erdős-Ko-Rado theorem, Combinatorica3(3–4) (1983), 341–349.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    P. Frankl and Z. Füredi: Extremal problems whose solutions are the blowups of the small Witt-designs, J. Combin. Theory Ser. A52 (1989), 129–147.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    P. Frankl and V. Rödl: Lower bounds for Turán’s problem, Graphs Combin. 1(3) (1985), 213–216.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    Z. Füredi, D. Mubayi and O. Pikhurko: Quadruple systems with independent neighborhoods, J. Combin. Theory Ser. A115(8) (2008), 1552–1560.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    Z. Füredi, O. Pikhurko and M. Simonovits: On triple systems with independent neighborhoods, Combin. Probab. Comput. 14 (2005), 795–813.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    Z. Füredi, O. Pikhurko and M. Simonovits: The Turán density of the hypergraph {abc, ade, bde, cde}; Electron. J. Combin. 10 (2003), Research Paper 18, 7 pp. (electronic).

  11. [11]

    S. Janson, T. Łuczak and A. Rucinski: Random Graphs, Wiley, 2000.

  12. [12]

    G. Katona, T. Nemetz and M. Simonovits: On a problem of Turán in the theory of graphs, Mat. Lapok15 (1964), 228–238.

    MATH  MathSciNet  Google Scholar 

  13. [13]

    P. Keevash: A hypergraph regularity method for generalised Turán problems, Random Structures and Algorithms34 (2009), 123–164.

    MATH  Article  MathSciNet  Google Scholar 

  14. [14]

    P. Keevash and D. Mubayi: Stability results for cancellative hypergraphs, J. Combin. Theory Ser. B92 (2004), 163–175.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    P. Keevash and Y. Zhao: Codegree problems for projective geometries, J. Combin. Theory Ser. B97 (2007), 919–928.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    K. Kim and F. Roush: On a problem of Turán, in: Studies in pure mathematics, pp. 423–425, Birkhäuser, Basel-Boston, Mass., 1983.

    Google Scholar 

  17. [17]

    D. Mubayi: The co-degree density of the Fano plane, J. Combin. Theory Ser. B95(2) (2005), 333–337.

    MATH  Article  MathSciNet  Google Scholar 

  18. [18]

    D. Mubayi and V. Rödl: On the Turán number of triple systems, J. Combin. Theory Ser. A100(1) (2002), 136–152.

    MATH  Article  MathSciNet  Google Scholar 

  19. [19]

    D. Mubayi and Y. Zhao: Codegree density of hypergraphs, J. Combin. Theory Ser. A114(6) (2007), 1118–1132.

    MATH  Article  MathSciNet  Google Scholar 

  20. [20]

    W. L. Nicholson: On the normal approximation to the hypergeometric distribution, Ann. Math. Statist. 27 (1956), 471–483.

    MATH  Article  MathSciNet  Google Scholar 

  21. [21]

    O. Pikhurko: An exact Turán result for the generalized triangle, Combinatorica28(2) (2008), 187–208.

    MATH  Article  MathSciNet  Google Scholar 

  22. [22]

    V. Rödl, A. Ruciński and E. Szemerédi: A Dirac-type theorem for 3-uniform hypergraphs, Combinatorics, Probability and Computing15(1–2) (2006), 229–251.

    MATH  Article  MathSciNet  Google Scholar 

  23. [23]

    A. F. Sidorenko: The maximal number of edges in a homogeneous hypergraph containing no prohibited subgraphs, Math Notes41 (1987), 247–259. Translated from Mat. Zametki.

    MATH  MathSciNet  Google Scholar 

  24. [24]

    P. Turán: On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok48 (1941), 436–452.

    MATH  MathSciNet  Google Scholar 

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Correspondence to Tom Bohman.

Additional information

Research supported in part by NSF grant DMS-0701183.

Research supported in part by NSF grant CCR-0502793.

Research supported in part by NSF grant DMS-0653946.

Research supported in part by NSF grant DMS-0457512.

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Bohman, T., Frieze, A., Mubayi, D. et al. Hypergraphs with independent neighborhoods. Combinatorica 30, 277–293 (2010). https://doi.org/10.1007/s00493-010-2474-6

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Mathematics Subject Classification (2000)

  • 05D05
  • 05C65