An exact Turán result for the generalized triangle

Abstract

Let Σ k consist of all k-graphs with three edges D 1, D 2, D 3 such that |D 1D 2| = k − 1 and D 1 Δ D 2D 3. The exact value of the Turán function ex(n, Σ k ) was computed for k = 3 by Bollobás [Discrete Math. 8 (1974), 21–24] and for k = 4 by Sidorenko [Math Notes 41 (1987), 247–259].

Let the k-graph T k Σ k have edges

$$ \{ 1, \ldots ,k\} , \{ 1,2, \ldots ,k - 1,k + 1\} , and \{ k,k + 1, \ldots ,2k - 1\} . $$

Frankl and Füredi [J. Combin. Theory Ser. (A) 52 (1989), 129–147] conjectured that there is n 0 = n 0(k) such that ex(n, T k ) = ex(n, Σ k ) for all nn 0 and had previously proved this for k = 3 in [Combinatorica 3 (1983), 341–349]. Here we settle the case k = 4 of the conjecture.

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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.

    Article  MathSciNet  MATH  Google Scholar 

  2. [2]

    D. de Caen: Uniform hypergraphs with no block containing the symmetric difference of any two other blocks, Congres. Numer. 47 (1985), 249–253.

    Google Scholar 

  3. [3]

    G. Elek and B. Szegedy: Limits of hypergraphs, removal and regularity lemmas. A non-standard approach; submitted (2007), arXiv:0705.2179.

  4. [4]

    P. Erdős and M. Simonovits: Supersaturated graphs and hypergraphs, Combinatorica 3(2) (1983), 181–192.

    Article  MathSciNet  Google Scholar 

  5. [5]

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

    Article  MathSciNet  MATH  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. (A) 52 (1989), 129–147.

    Article  MathSciNet  MATH  Google Scholar 

  7. [7]

    Z. Füredi and M. Simonovits: Triple systems not containing a Fano configuration, Combin. Prob. Computing 14 (2005), 467–488.

    Article  MATH  Google Scholar 

  8. [8]

    W. T. Gowers: Hypergraph regularity and the multidimensional Szemerédi’s theorem, submitted (2007), arXiv:0710.3032.

  9. [9]

    G. O. H. Katona: Extremal problems for hypergraphs, Combinatorics vol. 56, Math. Cent. Tracts, 1974, pp. 13–42.

    Google Scholar 

  10. [10]

    G. O. H. Katona, T. Nemetz and M. Simonovits: On a graph problem of Turán (In Hungarian), Mat. Fiz. Lapok 15 (1964), 228–238.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    P. Keevash and D. Mubayi: Stability results for cancellative hypergraphs, J. Combin. Theory Ser. (B) 92 (2004), 163–175.

    Article  MathSciNet  MATH  Google Scholar 

  12. [12]

    P. Keevash and B. Sudakov: The Turán number of the Fano plane, Combinatorica 25(5) (2005), 561–574.

    Article  MathSciNet  MATH  Google Scholar 

  13. [13]

    P. Keevash and B. Sudakov: On a hypergraph Turán problem of Frankl, Combinatorica 25(6) (2005), 673–706.

    Article  MathSciNet  MATH  Google Scholar 

  14. [14]

    W. Mantel: Problem 28, Winkundige Opgaven 10 (1907), 60–61.

    Google Scholar 

  15. [15]

    D. Mubayi and O. Pikhurko: A new generalization of Mantel’s theorem to k-graphs, J. Combin. Theory Ser. (B) 97 (2007), 669–678.

    Article  MathSciNet  MATH  Google Scholar 

  16. [16]

    B. Nagle, V. Rödl and M. Schacht: The counting lemma for regular k-uniform hypergraphs, Random Struct. Algorithms 28 (2005), 113–179.

    Article  Google Scholar 

  17. [17]

    O. Pikhurko: Exact computation of the hypergraph Turán function for expanded complete 2-graphs, arXiv:math/0510227, accepted by J. Comb. Th. Ser. (B). The publication is suspended because of a disagreement over copyright, see http://www.math.cmu.edu/:_pikhurko/Copyright.html, 2005.

  18. [18]

    V. Rödl and J. Skokan: Regularity lemma for k-uniform hypergraphs, Random Struct. Algorithms 25 (2004), 1–42.

    Article  MATH  Google Scholar 

  19. [19]

    V. Rödl and J. Skokan: Applications of the regularity lemma for uniform hypergraphs, Random Struct. Algorithms 28 (2006), 180–194.

    Article  MATH  Google Scholar 

  20. [20]

    J. B. Shearer: A new construction for cancellative families of sets, Electronic J. Combin. 3 (1996), 3 pp.

  21. [21]

    A. F. Sidorenko: The maximal number of edges in a homogeneous hypergraph containing no prohibited subgraphs, Math Notes 41 (1987), 247–259, Translated from Mat. Zametki.

    MathSciNet  MATH  Google Scholar 

  22. [22]

    T. Tao: A variant of the hypergraph removal lemma, J. Combin. Theory Ser. (A) 113 (2006), 1257–1280.

    Article  MathSciNet  MATH  Google Scholar 

  23. [23]

    P. Turán: On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941), 436–452.

    MathSciNet  MATH  Google Scholar 

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Correspondence to Oleg Pikhurko.

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Reverts to public domain after 28 years from publication.

Partially supported by the National Science Foundation, Grant DMS-0457512.

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Pikhurko, O. An exact Turán result for the generalized triangle. Combinatorica 28, 187–208 (2008). https://doi.org/10.1007/s00493-008-2187-2

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Mathematics Subject Classification (2000)

  • 05D05
  • 05C35