Turán’s Theorem for the Fano Plane

Abstract

Confirming a conjecture of Vera T. Sós in a very strong sense, we give a complete solution to Turán's hypergraph problem for the Fano plane. That is we prove for n≥8 that among all 3-uniform hypergraphs on n vertices not containing the Fano plane there is indeed exactly one whose number of edges is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for n = 7 there is exactly one other extremal configuration with the same number of edges: the hypergraph arising from a clique of order 7 by removing all five edges containing a fixed pair of vertices.

For sufficiently large values n this was proved earlier by Füredi and Simonovits, and by Keevash and Sudakov, who utilised the stability method.

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

References

  1. [1]

    B. Bollobás: Extremal graph theory, Dover Publications, Inc., Mineola, NY, 2004.

    MATH  Google Scholar 

  2. [2]

    W. G. Brown,: On an open problem of Paul Turán concerning 3-graphs, Studies in pure mathematics, Birkhäuser, Basel (1983), p91–93.

    MATH  Google Scholar 

  3. [3]

    D. De Caen and Z. Füredi: The maximum size of 3-uniform hypergraphs not containing a Fano plane, . Combin. Theory Ser. B78 (2000), 274–276.

    MathSciNet  Article  Google Scholar 

  4. [4]

    P. Erdős: Paul Turán, 1910-1976: his work in graph theory, J. Graph Theory1, (1977), 97–101.

    MathSciNet  Article  Google Scholar 

  5. [5]

    P. Erdős, P. A. Hajnal, A., V. T. Sós and E. Szemerédi: More results on Ramsey-Turan type problems, Combinatorica 3 (1983), 69–81.

    Google Scholar 

  6. [6]

    Z. Füredi: Personal communication.

  7. [7]

    Z. Füredi and A. Kündgen: Turán problems for integer-weighted graphs, J. Graph Theory40 (2002), 195–225.

    MathSciNet  Article  Google Scholar 

  8. [8]

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

    MathSciNet  Article  Google Scholar 

  9. [9]

    Gy. Katona, T. Nemetz and M. Simonovits: On a problem of Turán in the theory of graphs, Mat. Lapok15 (1964), 228–238. (In Hungarian, with Russian and English summaries.)

    MathSciNet  MATH  Google Scholar 

  10. [10]

    P. Keevash and D. Mubayi: The Turán number of F3,3, Combin. Probab. Comput.21 (2012), 451–456.

    MathSciNet  Article  Google Scholar 

  11. [11]

    P. Keevash and B. Sudakov: The Turán number of the Fano plane, Combinatorica25 (2005), 561–574.

    MathSciNet  Article  Google Scholar 

  12. [12]

    A. V. Kostochka: A class of constructions for Turán's (3,4)-problem, Combinatorica2 (1982), 187–192.

    MathSciNet  Article  Google Scholar 

  13. [13]

    C. M. Lüders and Ch. Reiher: The Ramsey-Turan problem for cliques, Israel Journal of Mathematics230 (2019), 613–652.

    MathSciNet  Article  Google Scholar 

  14. [14]

    C. M. Lüders and Ch. Reiher: Weighted variants of the Andrasfai-Erdős-Sós Theorem, to appear in Journal of Combinatorics.

  15. [15]

    D. Mubayi and V. Rödl: On the Turán number of triple systems, J. Combin. Theory Ser. A100 (2002), 136–152.

    MathSciNet  Article  Google Scholar 

  16. [16]

    M. Pasch: Vorlesungenüber neuere Geometrie, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], second edition, B. G. Teubner, Leipzig und Berlin (1912). (In German).

  17. [17]

    A. A. Razborov: Flag algebras, J. Symbolic Logic72 (2007), 1239–1282.

    MathSciNet  Article  Google Scholar 

  18. [18]

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

    MathSciNet  Article  Google Scholar 

  19. [19]

    V. Rödl and A. Sidorenko: On the jumping constant conjecture for multigraphs, J. Combin. Theory Ser. A69 (1995), 347–357.

    MathSciNet  Article  Google Scholar 

  20. [20]

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

    MATH  Google Scholar 

  21. [21]

    V. T. Sós: Remarks on the connection of graph theory, finite geometry and block designs, Colloquio Internazionale sulle Teorie Combinatorie, Roma, 1973, Accad. Naz. Lincei, Rome, 223–233, 1976.

    Google Scholar 

  22. [22]

    P. Turán: Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok48 (1941), 436–452. (In Hungarian, with German summary.)

    MathSciNet  MATH  Google Scholar 

  23. [23]

    W. T. Tutte: The factorization of linear graphs, J. London Math. Soc.22 (1947), 107–111.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgement

We would like to thank Miklós Simonovits for sending us a copy of [21], Zoltán Füredi [6] for further information regarding the history of the problem, and the referees for a careful reading of this article.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Christian Reiher.

Additional information

The second author was supported by the European Research Council (ERC grant PEPCo 724903).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bellmann, L., Reiher, C. Turán’s Theorem for the Fano Plane. Combinatorica 39, 961–982 (2019). https://doi.org/10.1007/s00493-019-3981-8

Download citation

Mathematics Subject Classification (2010)

  • 05C65
  • 05D05