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.
Similar content being viewed by others
References
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.
W. G. Brown: On an open problem of Paul Turán concerning 3-graphs, in: Studies in pure mathematics, 91–93. Birkhäuser, Basel, 1983.
F. Chung and L. Lu: An upper bound for the Turán number t3(n,4), J. Comb. Theory, Ser. A 87 (1999), 381–389.
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).
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.
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.
D. G. Fon-Der-Flaass: On a method of construction of (3,4)-graphs, Mat. Zametki 44 (1988), 546–550.
Z. Füredi, O. Pikhurko and M. Simonovits: On triple systems with independent neighbourhoods, Combin. Probab. Comput. 14 (2005), 795–813.
Z. Füredi and M. Simonovits: Triple systems not containing a Fano configuration, Comb. Probab. Comput. 14 (2005), 467–484.
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.
P. Keevash and B. Sudakov: On a hypergraph Turán problem of Frankl, Combinatorica 25 (2005), 673–706.
P. Keevash and B. Sudakov: The Turáan number of the Fano plane, Combinatorica 25 (2005), 561–574.
A. V. Kostochka: A class of constructions for Turán’s (3, 4)-problem, Combinatorica 2 (1982), 187–192.
X. Liu and D. Mubayi: A hypergraph Turán problem with no stability, arXiv:1911.07969, 2019.
X. Liu and D. Mubayi: The feasible region of hypergraphs, J. Combin. Theory Ser. B 148 (2021), 23–59.
X. Liu, D. Mubayi and C. Reiher: Hypergraphs with many extremal configurations, arXiv:2102.02103, 2021.
D. Mubayi: A hypergraph extension of Turáan’s theorem, J. Combin. Theory Ser. B 96 (2006), 122–134.
D. Mubayi: Structure and stability of triangle-free set systems, Trans. Amer. Math. Soc. 359 (2007), 275–291.
D. Mubayi and O. Pikhurko: A new generalization of Mantel’s theorem to k-graphs, J. Comb. Theory, Ser. B 97 (2007), 669–678.
D. Mubayi, O. Pikhurko and B. Sudakov: Hypergraph Turáan problem: Some open questions, 2011.
O. Pikhurko: An exact Turaán result for the generalized triangle, Combinatorica 28 (2008), 187–208.
O. Pikhurko: Exact computation of the hypergraph Turán function for expanded complete 2-graphs, J. Comb. Theory, Ser. B 103 (2013), 220–225.
O. Pikhurko: On possible Turán densities, Israel J. Math. 201 (2014), 415–454.
A. Razborov: On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math. 24 (2010), 946–963.
A. Sidorenko: What we know and what we do not know about Turán numbers, Graphs Comb. 11 (1995), 179–199.
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.
P. Turán: On an extermal problem in graph theory, Mat. Fiz. Lapok 48 (1941), 436–452.
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
Corresponding author
Additional information
Research partially supported by NSF award DMS-1763317.