Israel Journal of Mathematics

, Volume 211, Issue 1, pp 349–366

A problem of Erdős and Sós on 3-graphs



We show that for every ɛ > 0 there exist δ > 0 and n0 ∈ ℕ such that every 3-uniform hypergraph on nn0 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 K4 as a subgraph, where K4 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Alon and J. Spencer, The Probabilistic Method, 3rd Edition, Wiley-Interscience in Descrete Mathematics and Optimization, Wiley, Hoboken, NJ, 2008.CrossRefMATHGoogle Scholar
  2. [2]
    R. Baber and J. Talbot, Hypergraphs do jump, Combinatorics, Probability and Computing 20 (2011), 161–171.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    R. Baber and J. Talbot, New Turán densities for 3-graphs, Electronic Journal of Combinatorics 19 (2012), #P22.MathSciNetMATHGoogle Scholar
  4. [4]
    V. Bhat and V. Rödl, Note on upper density of quasi-random hypergraphs, preprint.Google Scholar
  5. [5]
    B. Borchers, CSDP, a C library for semidefinite programming, Optimization Methods and Software 11/12 (1999), 613–623.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    G. Elek and B. Szegedy, A measure theoretic approach to the theory of dense hypergraphs, preprint.Google Scholar
  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.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    P. Erdős and V. T. Sós, On Ramsey-Turán type theorems for hypergraphs, Combinatorica 2 (1982), 289–295.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle Scholar
  14. [14]
    V. Falgas-Ravry, O. Pikhurko and E. R. Vaughan, The codegree density of K - 4, in preparation.Google Scholar
  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.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    V. Falgas-Ravry and E. R. Vaughan, Turán H-densities for 3-graphs, Electronic Journal of Combinatorics 19 (2012), #P40.MathSciNetMATHGoogle Scholar
  17. [17]
    P. Frankl and Z. Füredi, An exact result for 3-graphs, Discrete Mathematics 50 (1984), 323–328.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    P. Frankl and V. Rödl, Some Ramsey-Turán type results for hypergraphs, Combinatorica 8 (1988), 323–332.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    L. Lovász and B. Szegedy, Limits of dense graph sequences, Journal of Combinatorial Theory, Series B 96 (2006), 933–957.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    A. Razborov, Flag algebras, Journal of Symbolic Logic 72 (2007), 1239–1282.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    A. Razborov, On 3-hypergraphs with forbidden 4-vertex configurations, SIAM Journal on Discrete Mathematics 24 (2010), 946–963.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    A. Razborov, On Turánś (3, 4)-problem with forbidden configurations, preprint.Google Scholar
  25. [25]
    V. Rödl, On universality of graphs with uniformly distributed edges, Discrete Mathematics 59 (1986), 125–134.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    V. Rödl, private communication, 2013.Google Scholar
  27. [27]
    W. A. Stein et al., Sage Mathematics Software (Version 5.4.1), The Sage Development Team, 2012,
  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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    J. Volec, Analytic methods in combinatorics, Ph.D. thesis, University of Warwick and Université Paris Diderot (2014).Google Scholar

Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Department of MathematicsETHZurichSwitzerland
  2. 2.Mathematics InstituteDIMAP and Department of Computer Science University of WarwickCoventryUK

Personalised recommendations