Skip to main content

A Hypergraph Turán Problem with No Stability

Abstract

A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the tetrahedron exhibits this phenomenon assuming Turán’s conjecture.

Our main result is to construct a finite family of triple systems \({\cal M}\), determine its Turán number, and prove that there are two near-extremal \({\cal M}\)-free constructions that are far from each other in edit-distance. This is the first extremal result for a hypergraph family that fails to have a corresponding stability theorem.

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

References

  1. A. Brandt, D. Irwin and T. Jiang: Stability and Turán numbers of a class of hypergraphs via Lagrangians, Combin. Probab. Comput. 26 (2017), 367–405.

    MathSciNet  Article  Google Scholar 

  2. W. G. Brown: On an open problem of Paul Turán concerning 3-graphs, in: Studies in pure mathematics, 91–93. Birkhäuser, Basel, 1983.

    Chapter  Google Scholar 

  3. F. Chung and L. Lu: An upper bound for the Turán number t3(n,4), J. Comb. Theory, Ser. A 87 (1999), 381–389.

    Article  Google Scholar 

  4. D. de Caen: On upper bounds for 3-graphs without tetrahedra, volume 62, 193–202, 1988, Seventeenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1987).

  5. D. De Caen and Z. Füredi: The maximum size of 3-uniform hypergraphs not containing a Fano plane, J. Combin. Theory Ser. B 78 (2000), 274–276.

    MathSciNet  Article  Google Scholar 

  6. V. Falgas-Ravry and E. R. Vaughan: Applications of the semi-definite method to the Turán density problem for 3-graphs, Combin. Probab. Comput. 22 (2013), 21–54.

    MathSciNet  Article  Google Scholar 

  7. D. G. Fon-Der-Flaass: On a method of construction of (3,4)-graphs, Mat. Zametki 44 (1988), 546–550.

    MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  9. Z. Füredi and M. Simonovits: Triple systems not containing a Fano configuration, Comb. Probab. Comput. 14 (2005), 467–484.

    MathSciNet  Article  Google Scholar 

  10. P. Keevash: Hypergraph Turán problems, in: Surveys in combinatorics 2011, volume 392 of London Math. Soc. Lecture Note Ser., 83–139, Cambridge Univ. Press, Cambridge, 2011.

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  13. A. V. Kostochka: A class of constructions for Turán’s (3, 4)-problem, Combinatorica 2 (1982), 187–192.

    MathSciNet  Article  Google Scholar 

  14. X. Liu and D. Mubayi: A hypergraph Turán problem with no stability, arXiv:1911.07969, 2019.

  15. X. Liu and D. Mubayi: The feasible region of hypergraphs, J. Combin. Theory Ser. B 148 (2021), 23–59.

    MathSciNet  Article  Google Scholar 

  16. X. Liu, D. Mubayi and C. Reiher: Hypergraphs with many extremal configurations, arXiv:2102.02103, 2021.

  17. D. Mubayi: A hypergraph extension of Turáan’s theorem, J. Combin. Theory Ser. B 96 (2006), 122–134.

    MathSciNet  Article  Google Scholar 

  18. D. Mubayi: Structure and stability of triangle-free set systems, Trans. Amer. Math. Soc. 359 (2007), 275–291.

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  20. D. Mubayi, O. Pikhurko and B. Sudakov: Hypergraph Turáan problem: Some open questions, 2011.

  21. O. Pikhurko: An exact Turaán result for the generalized triangle, Combinatorica 28 (2008), 187–208.

    MathSciNet  Article  Google Scholar 

  22. O. Pikhurko: Exact computation of the hypergraph Turán function for expanded complete 2-graphs, J. Comb. Theory, Ser. B 103 (2013), 220–225.

    Article  Google Scholar 

  23. O. Pikhurko: On possible Turán densities, Israel J. Math. 201 (2014), 415–454.

    MathSciNet  Article  Google Scholar 

  24. A. Razborov: On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math. 24 (2010), 946–963.

    MathSciNet  Article  Google Scholar 

  25. A. Sidorenko: What we know and what we do not know about Turán numbers, Graphs Comb. 11 (1995), 179–199.

    Article  Google Scholar 

  26. M. Simonovits: A method for solving extremal problems in graph theory, stability problems, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), 279–319, Academic Press, New York, 1968.

    Google Scholar 

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

    MathSciNet  Google Scholar 

Download references

Acknowledgment

We are very grateful to all the referees for their many helpful comments. In particular, for the suggestion of using the result in [8] which substantially shortened the presentation and for the cleaner and shorter proofs of some technical statements (Lemma 3.3 and Claim 4.13).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dhruv Mubayi.

Additional information

Research partially supported by NSF award DMS-1763317.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Mubayi, D. A Hypergraph Turán Problem with No Stability. Combinatorica 42, 433–462 (2022). https://doi.org/10.1007/s00493-021-4561-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-021-4561-2

Mathematics Subject Classification (2010)

  • 05C35
  • 05D05