Resilient Hypergraphs with Fixed Matching Number

Abstract

Let H be a hypergraph of rank k, that is, |H| ≦ k for all HH. Let ν(H) denote the matching number, the maximum number of pairwise disjoint edges in H. For a vertex x let H(x̄) be the hypergraph consisting of the edges HH with xH. If ν(H(x̄)) = ν(H) for all vertices, H is called resilient. The main result is the complete determination of the maximum number of 2-element sets in a resilient hypergraph with matching number s. For k=3 it is \(\left( {\begin{array}{*{20}c} {2s + 1} \\ 2 \\ \end{array} } \right)\) while for k ≧ 4 the formula is \(k \cdot \left( {\begin{array}{*{20}c} {2s + 1} \\ 2 \\ \end{array} } \right)\). The results are used to obtain a stability theorem for k-uniform hypergraphs with given matching number.

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

References

  1. [1]

    D. Conlon and V. Rödl: Personal Communication, 2015.

  2. [2]

    P. Erdős: A problem on independent r-tuples, Ann. Univ. Sci. Budapest 8 (1965), 93–95.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    P. Erdős, C. Ko and R. Rado: Intersection theorems for systems of finite sets, Quart. J. Math. Oxford 12 (1961), 313–320.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    P. Erdős and L. Lovász: Problems and results on 3-chromatic hypergraphs and some related questions, in: Infinite and Finite Sets, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1974, 609–627.

    Google Scholar 

  5. [5]

    P. Erdős and R. Rado: Intersection theorems for systems of sets, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    P. Frankl: On the maximum number of edges in a hypergraph with given matching number, Discrete Applied Mathematics (2016). arXiv:1205.6847 (May, 30, 2012).

    Google Scholar 

  7. [7]

    P. Frankl: Improved bounds for Erdős’ matching conjecture, J. Comb. Theory, Ser. A 120(5) (2013), 1068–1072.

    Article  MATH  Google Scholar 

  8. [8]

    P. Frankl: On intersecting families of finite sets, Bull. Austral. Math. Soc. 21 (1980), 363–372.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    P. Frankl, K. Ota and N. Tokushige: Uniform intersecting families with covering number four, J. Comb. Theory, Ser A 71 (1995), 127–145.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    P. Frankl, K. Ota and N. Tokushige: Covers in uniform intersecting families and a counterexample to a conjecture of Lovász, J. Comb. Theory, Ser A 74 (1996), 33–42.

    Article  MATH  Google Scholar 

  11. [11]

    M. Furuya and M. Takatou: Covers in 5-uniform intersecting families with covering number three, Australas. J. Combin. 55 (2013), 249–262.

    MathSciNet  MATH  Google Scholar 

  12. [12]

    M. Furuya and M. Takatou: The number of covers in intersecting families with covering number three, preprint.

  13. [13]

    A. J. W. Hilton and E. C. Milner: Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford 18 (1967), 369–384.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    A. Kostochka and D. Mubayi: The structure of large intersecting families, manuscript, 2016.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Peter Frankl.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Frankl, P. Resilient Hypergraphs with Fixed Matching Number. Combinatorica 38, 1079–1094 (2018). https://doi.org/10.1007/s00493-016-3579-3

Download citation

Mathematics Subject Classification (2000)

  • 05D05
  • 05C65