A Turán Theorem for Extensions Via an Erdős-Ko-Rado Theorem for Lagrangians


The extension of an r-uniform hypergraph G is obtained from it by adding for every pair of vertices of G, which is not covered by an edge in G, an extra edge containing this pair and (r−2) new vertices. In this paper we determine the Turán number of the extension of an r-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Turán number for r=3. As the key ingredient of the proof we show that the Lagrangian of intersecting r-graphs is maximized by principally intersecting r-graphs for r≥4.

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  1. [1]

    R. Ahlswede and L. H. Khachatrian: The complete intersection theorem for systems of finite sets, European J. Combin.18 (1997), 125–136.

    MathSciNet  Article  Google Scholar 

  2. [2]

    A. Brandt, D. Irwin and T. Jiang: Stability and Tur_an numbers of a class ofhypergraphs via Lagrangians, Combinatorics, Probability, and Computing2 (2017), 367–405.

    Article  Google Scholar 

  3. [3]

    P. Frankl: The shifting technique in extremal set theory, Surveys in combinatorics1987 (New Cross, 1987), London Math. Soc. Lecture Note Ser., vol. 123, CambridgeUniv. Press, Cambridge, 1987, 81–110.

    Google Scholar 

  4. [4]

    P. Frankl and V. Rődl: Hypergraphs do not jump, Journal of Symbolic Logic4(1984), 149–159.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    E. Friedgut: On the measure of intersecting families, uniqueness and stability, Combinatorica28 (2008), 503–528.

    MathSciNet  Article  Google Scholar 

  6. [6]

    D. Hefetz and P. Keevash: A hypergraph Turán theorem via Lagrangians of intersecting families, J. Combin. Theory Ser. A120 (2013), 2020–2038.

    MathSciNet  Article  Google Scholar 

  7. [7]

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

    MathSciNet  MATH  Google Scholar 

  8. [8]

    T. S. Motzkin and E. G. Straus: Maxima for graphs and a new proof of a theoremof Turán, Canad. J. Math. (1965), 533–540.

    Google Scholar 

  9. [9]

    S. Norin and L. Yepremyan: On Turán numbers of extensions, arXiv:1510.04689, 2015.

    Google Scholar 

  10. [10]

    S. Norin and L. Yepremyan: Turán number of generalized triangles, J. Combin.Theory Ser. A146 (2017), 312–343.

    MathSciNet  Article  Google Scholar 

  11. [11]

    O. Pikhurko: Exact computation of the hypergraph Turán function for expandedcomplete 2-graphs, J. Combin. Theory Ser. B103 (2013), 220–225.

    MathSciNet  Article  Google Scholar 

  12. [12]

    A. F. Sidorenko: The maximal number of edges in a homogeneous hypergraphcontaining no prohibited subgraphs, Mathematical notes of the Academy of Sciencesof the USSR41 (1987), 247–259.

    Article  Google Scholar 

  13. [13]

    B. Wu, Y. Peng and P. Chen: On a conjecture of Hefetz and Keevash on lagrangiansof intersecting hypergraphs and Turán numbers, arXiv:1701.06126, 2017.

    Google Scholar 

  14. [14]

    L. Yepremyan: Local stability method for hypergraph Turán problems, Ph.D. thesis, McGill University, 2016.

    Google Scholar 

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Some of this research was performed in Summer 2015, when the first author was an undergraduate student at McGill University supported by an NSERC USRA scholarship.

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Correspondence to Liana Yepremyan.

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Watts, A.B., Norin, S. & Yepremyan, L. A Turán Theorem for Extensions Via an Erdős-Ko-Rado Theorem for Lagrangians. Combinatorica 39, 1149–1171 (2019). https://doi.org/10.1007/s00493-019-3831-8

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Mathematics Subject Classification (2010)

  • 05C35
  • 05C65
  • 05D05
  • 04D40