An Erdős–Ko–Rado type theorem for subgraphs of perfect matchings

Abstract

Let $M_k$ be a 2n-vertex graph with n pairwise disjoint edges and let $H^{(p,s)}\left(n\right)$ be the family of subsets of $V\left(M_n\right)$ that span exactly p edges and s isolated vertices. We prove that for $n\ge 2p+s$ this family has the Erdős–Ko–Rado property: the size of the largest intersecting family is equal to the number of sets containing a fixed vertex. The bound $n\ge 2p+s$ is the best possible, improving a recent theorem with $n\ge 2p+2s$ by Fuentes and Kamat.