On intersecting families of subgraphs of perfect matchings

Abstract

The seminal Erdős–Ko–Rado (EKR) theorem states that if $F$ is a family of k-subsets of an n-element set X for $k\le n/2$ such that every pair of subsets in $F$ has a nonempty intersection, then $F$ can be no bigger than the trivially intersecting family obtained by including all k-subsets of X that contain a fixed element $x\in X$. This family is called the star centered at x. In this paper, we formulate and prove an EKR theorem for intersecting families of subgraphs of the perfect matching graph. This can be considered a generalization not only of the aforementioned EKR theorem but also of a signed variant of it, first stated by Meyer [9], and proved separately by Deza–Frankl [3] and Bollobás–Leader [1]. The proof of our main theorem relies on a novel extension of Katona's beautiful cycle method.