Advertisement

A problem of Erdős and Sós on 3-graphs

Conference paper
Part of the CRM Series book series (PSNS, volume 16)

Abstract

We show that for every ε > 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 k ≥ δn, induces at least (1/4 + ɛ) 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós and K. Vesztergombi, sequences of dense graphs I: Subgraph frequencies, metric properties and testing, Advances in Mathematics 219 (2008), 1801–1851.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    P. Erdős, Problems and results on graphs and hypergraphs: Similarities and differences, In: “Mathematics of Ramsey Theory”, J. J. Nešetřil and V. Rödl (eds.), Springer-Verlag, 1990, 223–233.Google Scholar
  4. [4]
    P. Erdős and A. Hajnal, On Ramsey like theorems. Problems and results, In: “Combinatorics: being the proceedings of the Conference on Combinatorial Mathematics held at the Mathematical Institute”, Oxford, 1972, Southend-on-Sea: Institute of Mathematics and Its Applications, 1972, 123–140.Google Scholar
  5. [5]
    P. Erdős and M. Simonovits, An extremal graph problem, Acta Mathematica Academiae Scientiarum Hungaricae 22 (1971), 275–282.CrossRefMathSciNetGoogle Scholar
  6. [6]
    P. Erdős and V. T. Sós, On Ramsey-Turén type theorems for hypergraphs, Combinatorica 2 (1982), 289–295.CrossRefMathSciNetGoogle Scholar
  7. [7]
    P. Erdős and A. H. Stone, On the structure of linear graphs, Bulletin of the American Mathematical Society 52 (1946), 1089–1091.Google Scholar
  8. [8]
    P. Frankl and V. Rödl, Some Ramsey-Turán type results for hypergraphs, Combinatorica 8 (1988), 323–332.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    L. Lovász and B. Szegedy, Limits of dense graph sequences, Journal of Combinatorial Theory, Series B 96 (2006), 933–957.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    A. Razborov, Flag algebras, Journal of Symbolic Logic, 72 (2007), 1239–1282.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    V. Rödl, On universality of graphs with uniformly distributed edges, Discrete Mathematics 59 (1986), 125–134.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    M. Simonovits and V. T. Sós, Ramsey-Turán theory, Discrete Mathematics 229 (2001), 293–340.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    V. T. Sós, On extremal problems in graph theory, In: “Proceedings of the Calgary International Conference on Combinatorial Structures and their Application”, 1969, 407–410.Google Scholar
  14. [14]
    P. Turán, Eine Extremalaufgabe aus der Graphentheorie (in Hungarian), Matematikai és Fizikai Lapok 48 (1941), 436–452, see also: On the theory of graphs, Colloquium Mathematicum 3 (1954), 19–30.Google Scholar

Copyright information

© Scuola Normale Superiore Pisa 2013

Authors and Affiliations

  1. 1.Mathematics Institute and DIMAPUniversity of WarwickCoventryUK
  2. 2.Mathematics Institute, DIMAP and Department of Computer ScienceUniversity of WarwickCoventryUK

Personalised recommendations