Matching criticality in intersecting hypergraphs

A matching in a hypergraph $H$ is a set of pairwise vertex disjoint edges in $H$ and the matching number of $H$ is the maximum cardinality of a matching in $H$. A transversal in $H$ is a subset of vertices in $H$ that has a nonempty intersection with every edge of $H$. The transversal number $\tau(H)$ of $H$ is the minimum cardinality of a transversal in $H$. A hypergraph $H$ is an intersecting hypergraph if every two distinct edges of $H$ have a non-empty intersection. Equivalently, $H$ is an intersecting hypergraph if and only if it has matching number one. In this paper we study the extremal behavior of matching critical intersecting hypergraphs. We partly solve an open problem on matching critical intersecting hypergraphs posed by Henning and Yeo. We also prove a strengthening of the result for intersecting $r$-uniform hypergraphs.