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A Hypergraph Turán Problem with No Stability

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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.

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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).

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL, 60607, USA

    Xizhi Liu & Dhruv Mubayi

Authors
  1. Xizhi Liu
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  2. Dhruv Mubayi
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Correspondence to Dhruv Mubayi.

Additional information

Research partially supported by NSF award DMS-1763317.

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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

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

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