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

Let M k be a 2 n -vertex graph with n pairwise disjoint edges and let H ( p , s ) ( n ) be the family of subsets of V ( M n ) that span exactly p edges and s isolated vertices. We prove that for n ≥ 2 p + 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 ≥ 2 p + s is the best possible, improving a recent theorem with n ≥ 2 p + 2 s by Fuentes and Kamat.