On intersecting families of subgraphs of perfect matchings

TL;DR

This paper extends the Erd ext{"o}s--Ko--Rado theorem to intersecting families of subgraphs within perfect matching graphs, generalizing previous results and employing a novel cycle method for proof.

Contribution

It formulates and proves a new EKR-type theorem for subgraphs of perfect matchings, broadening the scope of classical intersection theorems.

Findings

Established an EKR theorem for subgraphs of perfect matchings.

Generalized previous EKR results and signed variants.

Introduced a new cycle method for the proof.

Abstract

The seminal Erd\H{o}s--Ko--Rado (EKR) theorem states that if $\mathcal{F}$ is a family of $k$-subsets of an $n$-element set $X$ for $k\leq n/2$ such that every pair of subsets in $\mathcal{F}$ has a nonempty intersection, then $\mathcal{F}$ 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, the graph consisting of $n$ disjoint edges. 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 (1974), and proved separately by Deza--Frankl (1983) and Bollob\'as--Leader (1997). The proof of our main theorem relies on a novel extension of Katona's beautiful cycle method.