A problem of Erdős and Sós on 3-graphs

Abstract

We show that for every ɛ > 0 there exist δ > 0 and n 0 ∈ ℕ such that every 3-uniform hypergraph on nn 0 vertices with the property that every k-vertex subset, where kδn, induces at least \(\left( {\frac{1} {2} + \varepsilon } \right)\left( {\begin{array}{*{20}c} k \\ 3 \\ \end{array} } \right)\) edges, contains K 4 as a subgraph, where K 4 is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdős and Sós. The constant 1/4 is the best possible.

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References

  1. [1]

    N. Alon and J. Spencer, The Probabilistic Method, 3rd Edition, Wiley-Interscience in Descrete Mathematics and Optimization, Wiley, Hoboken, NJ, 2008.

    Google Scholar 

  2. [2]

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

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    R. Baber and J. Talbot, New Turán densities for 3-graphs, Electronic Journal of Combinatorics 19 (2012), #P22.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    V. Bhat and V. Rödl, Note on upper density of quasi-random hypergraphs, preprint.

  5. [5]

    B. Borchers, CSDP, a C library for semidefinite programming, Optimization Methods and Software 11/12 (1999), 613–623.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós and K. Vesztergombi, Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing, Advances in Mathematics 219 (2008), 1801–1851.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós and K. Vesztergombi, Convergent sequences of dense graphs II: Multiway cuts and statistical physics, Annals of Mathematics 176 (2012), 151–219.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    G. Elek and B. Szegedy, A measure theoretic approach to the theory of dense hypergraphs, preprint.

  9. [9]

    P. Erdős, Problems and results on graphs and hypergraphs: Similarities and differences, in Mathematics of Ramsey Theory, Algorithms and Combinatorics, Vol. 5, Springer- Verlag, Berlin, 1990, pp. 223–233.

    Google Scholar 

  10. [10]

    P. Erdős and A. Hajnal, On Ramsey like theorems. Problems and results, in Combinatorics (Proceedings of the Conference on Combinatorial Mathematics held at the Mathematical Institute, Oxford, 1972), Institute of Mathematics and Its Applications, Southend-on-Sea, 1972, pp. 123–140.

    Google Scholar 

  11. [11]

    P. Erdős and M. Simonovits, An extremal graph problem, Acta Mathematica Academiae Scientiarum Hungaricae 22 (1971), 275–282.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

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

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    P. Erdős and A. H. Stone, On the structure of linear graphs, Bulletin of the American Mathematical Society 52 (1946), 1089–1091.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    V. Falgas-Ravry, O. Pikhurko and E. R. Vaughan, The codegree density of K - 4, in preparation.

  15. [15]

    V. Falgas-Ravry and E. R. Vaughan, Applications of the semi-definite method to the Turán density problem for 3-graphs, Combinatorics, Probability and Computing 22 (2013), 21–54.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    V. Falgas-Ravry and E. R. Vaughan, Turán H-densities for 3-graphs, Electronic Journal of Combinatorics 19 (2012), #P40.

    MathSciNet  MATH  Google Scholar 

  17. [17]

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

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    P. Frankl and V. Rödl, Some Ramsey-Turán type results for hypergraphs, Combinatorica 8 (1988), 323–332.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    L. Lovász and B. Szegedy, Limits of dense graph sequences, Journal of Combinatorial Theory, Series B 96 (2006), 933–957.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    O. Pikhurko, The minimum size of 3-graphs without a 4-set spanning no or exactly three edges, European Journal of Combinatorics 32 (2011), 1142–1155.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

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

    MathSciNet  Article  MATH  Google Scholar 

  22. [22]

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

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    A. Razborov, On the Fon-der-Flaass interpretation of extremal examples for Turánś (3, 4)-problem, Proceedings of the Steklov Institute of Mathematics 274 (2011), 247–266.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    A. Razborov, On Turánś (3, 4)-problem with forbidden configurations, preprint.

  25. [25]

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

    MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    V. Rödl, private communication, 2013.

  27. [27]

    W. A. Stein et al., Sage Mathematics Software (Version 5.4.1), The Sage Development Team, 2012, http://www.sagemath.org.

  28. [28]

    M. Simonovits, A method for solving extremal problems in graph theory, stability problems, in Theory of Graphs (Proceedings of the Colloquium held at Tihany, 1966), Academic Press, New York, 1968, pp. 279–319.

    Google Scholar 

  29. [29]

    M. Simonovits and V. T. Sós, Ramsey-Turán theory, Discrete Mathematics 229 (2001), 293–340.

    MathSciNet  Article  MATH  Google Scholar 

  30. [30]

    V. T. Sós, On extremal problems in graph theory, in Combinatorial Structures and their Applications (Proceedings of the Calgary International Conference, Calgary, Alta., 1969, Gordon and Breach, New York, 1970, pp. 407–410.

    Google Scholar 

  31. [31]

    P. Turán, Eine Extremalaufgabe aus der Graphentheorie (in Hungarian), Matematikaiés Fizikai Lapok 48 (1941), 436–452; see also: On the theory of graphs, Colloquium Mathematicum 3 (1954), 19–30.

    Google Scholar 

  32. [32]

    A. Tychonoff, Über die topologische Erweiterung von Räumen (in German), Mathematische Annalen 102 (1930), 544–561.

    MathSciNet  Article  MATH  Google Scholar 

  33. [33]

    J. Volec, Analytic methods in combinatorics, Ph.D. thesis, University of Warwick and Université Paris Diderot (2014).

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Correspondence to Roman Glebov.

Additional information

The work leading to this paper has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259385. The first author was also supported by DFG within the research training group “Methods for Discrete Structures”. The third author was also supported by the student grant GAUK 601812.

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Glebov, R., Král’, D. & Volec, J. A problem of Erdős and Sós on 3-graphs. Isr. J. Math. 211, 349–366 (2016). https://doi.org/10.1007/s11856-015-1267-4

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Keywords

  • Convergent Sequence
  • Edge Density
  • Dense Graph
  • Extremal Graph
  • Subgraph Frequency