Resilient Hypergraphs with Fixed Matching Number

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(x) be the hypergraph consisting of the edges H ∈ H with x ∉ H. If ν(H(x)) = 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.