Skip to main content
SpringerLink
Log in

Hypergraphs with minimum positive uniform Turán density

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Reiher, Rödl and Schacht showed that the uniform Turán density of every 3-uniform hypergraph is either 0 or at least 1/27, and asked whether there exist 3-uniform hypergraphs with uniform Turán density equal or arbitrarily close to 1/27. We construct 3-uniform hypergraphs with uniform Turán density equal to 1/27.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Baber and J. Talbot, Hypergraphs do jump, Combinatorics, Probability and Computing 20 (2011), 161–171.

    Article  MathSciNet  Google Scholar 

  2. M. Bucić, J. W. Cooper, D. Kráľ, S. Mohr and D. Munhá Correia, Uniform Turán density of cycles, Transactions of the American Mathematical Society 376 (2023), 4765–4809.

    Article  MathSciNet  Google Scholar 

  3. P. Erdős, Problems and results on graphs and hypergraphs: similarities and differences, in Mathematics of Ramsey Theory, Algorithms and Combinatorics, Vol. 5, Springer, Berlin–Heidelberg, 1990, pp. 12–28.

    Chapter  Google Scholar 

  4. P. Erdős and M. Simonovits, A limit theorem in graph theory, Studia Scientiarum Mathematicarum Hungarica 1 (1966), 51–57.

    MathSciNet  Google Scholar 

  5. P. Erdős and V. T. Sös, On Ramsey–Turán type theorems for hypergraphs, Combinatorica 2 (1982), 289–295.

    Article  MathSciNet  Google Scholar 

  6. P. Erdős and A. H. Stone, On the structure of linear graphs, Bulletin of the American Mathematical Society 52 (1946), 1087–1091.

    Article  MathSciNet  Google Scholar 

  7. P. Frankl and Z. Füredi, An exact result for 3-graphs, Discrete Mathematics 50 (1984), 323–328.

    Article  MathSciNet  Google Scholar 

  8. Z. Füredi, A proof of the stability of extremal graphs, Simonovits’ stability from Szemerédi’s regularity, Journal of Combinatorial Theory. Series B 115 (2015), 66–71.

    Article  MathSciNet  Google Scholar 

  9. R. Glebov, J. Volec and D. Kráľ, A problem of Erdős and Sós on 3-graphs, Israel Journal of Mathematics 211 (2016), 349–366.

    Article  MathSciNet  Google Scholar 

  10. G. Katona, T. Nemetz and M. Simonovits, On a problem of Turán in the theory of graphs, Matematikai Lapok 15 (1964), 228–238.

    MathSciNet  Google Scholar 

  11. P. Keevash, Hypergraph Turán problems, in Surveys in Combinatorics 2011, London Mathematical Society Lecture Note Series, Vol. 392, Cambridge University Press, Cambridge, 2011, pp. 83–139.

    Chapter  Google Scholar 

  12. W. Mantel, Problem 28, Wiskundige Opgaven 10 (1907), 60–61.

    Google Scholar 

  13. F. P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society 2 (1930), 264–286.

    Article  MathSciNet  Google Scholar 

  14. A. A. Razborov, Flag algebras, Journal of Symbolic Logic 72 (2007), 1239–1282.

    Article  MathSciNet  Google Scholar 

  15. A. A. Razborov, On 3-hypergraphs with forbidden 4-vertex configurations, SIAM Journal on Discrete Mathematics 24 (2010), 946–963.

    Article  MathSciNet  Google Scholar 

  16. C. Reiher, Extremal problems in uniformly dense hypergraphs, European Journal of Combinatorics 88 (2020), Article no. 103117.

  17. C. Reiher, V. Rüdl and M. Schacht, Embedding tetrahedra into quasirandom hypergraphs, Journal of Combinatorial Theory. Series B 121 (2016), 229–247.

    Article  MathSciNet  Google Scholar 

  18. C. Reiher, V. Rödl and M. Schacht, Hypergraphs with vanishing Turán density in uniformly dense hypergraphs, Journal of the London Mathematical Society 97 (2018), 77–97.

    Article  MathSciNet  Google Scholar 

  19. C. Reiher, V. Rödl and M. Schacht, On a generalisation of Mantel’s theorem to uniformly dense hypergraphs, International Mathematics Research Notices 2018 (2018), 4899–4941.

    Article  MathSciNet  Google Scholar 

  20. C. Reiher, V. Rödl and M. Schacht, On a Turán problem in weakly quasirandom 3-uniform hypergraphs, Journal of the European Mathematical Society 20 (2018), 1139–1159.

    Article  MathSciNet  Google Scholar 

  21. C. Reiher, V. Rödl and M. Schacht, Some remarks on π∧, in Connections in Discrete Mathematics Cambridge University Press, cambridge, 2018, pp. 214–239.

    Chapter  Google Scholar 

  22. V. Rödl, On universality of graphs with uniformly distributed edges, Discrete Mathematics 59 (1986), 125–134.

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  24. P. Turán, Eine Extremalaufgabe aus der Graphentheorie, Matematikai és Fizikai Lapok 48 (1941), 436–452.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Jacob Cooper for his comments on the topics covered in this paper. We would also like to thank the anonymous referee for their careful review of the manuscript and for their many helpful comments.

Author information

Authors and Affiliations

  1. Faculty of Informatics, Masaryk University, Botanická 68A, 602 00, Brno, Czech Republic

    Frederik Garbe, Daniel Král’ & Ander Lamaison

Authors
  1. Frederik Garbe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Král’
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ander Lamaison
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frederik Garbe.

Additional information

This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648509). This publication reflects only its authors’ view; the European Research Council Executive Agency is not responsible for any use that may be made of the information it contains. The authors were also supported by the MUNI Award in Science and Humanities of the Grant Agency of Masaryk University.

An extended abstract containing the results presented in this paper has appeared in the proceedings of EuroComb’21.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garbe, F., Král’, D. & Lamaison, A. Hypergraphs with minimum positive uniform Turán density. Isr. J. Math. 259, 701–726 (2024). https://doi.org/10.1007/s11856-023-2554-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-023-2554-0

Search

Navigation