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Combinatorica

, Volume 38, Issue 5, pp 1079–1094 | Cite as

Resilient Hypergraphs with Fixed Matching Number

  • Peter Frankl
Original Paper
  • 32 Downloads

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.

Mathematics Subject Classification (2000)

05D05 05C65 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Alfréd Rényi InstituteBudapestHungary

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