Abstract
Let H be a hypergraph of rank k, that is, |H| ≦ k for all H ∈ H. 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 H ∈ H with x ∉ H. 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.
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References
D. Conlon and V. Rödl: Personal Communication, 2015.
P. Erdős: A problem on independent r-tuples, Ann. Univ. Sci. Budapest 8 (1965), 93–95.
P. Erdős, C. Ko and R. Rado: Intersection theorems for systems of finite sets, Quart. J. Math. Oxford 12 (1961), 313–320.
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.
P. Erdős and R. Rado: Intersection theorems for systems of sets, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90.
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).
P. Frankl: Improved bounds for Erdős’ matching conjecture, J. Comb. Theory, Ser. A 120(5) (2013), 1068–1072.
P. Frankl: On intersecting families of finite sets, Bull. Austral. Math. Soc. 21 (1980), 363–372.
P. Frankl, K. Ota and N. Tokushige: Uniform intersecting families with covering number four, J. Comb. Theory, Ser A 71 (1995), 127–145.
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.
M. Furuya and M. Takatou: Covers in 5-uniform intersecting families with covering number three, Australas. J. Combin. 55 (2013), 249–262.
M. Furuya and M. Takatou: The number of covers in intersecting families with covering number three, preprint.
A. J. W. Hilton and E. C. Milner: Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford 18 (1967), 369–384.
A. Kostochka and D. Mubayi: The structure of large intersecting families, manuscript, 2016.
