Stability for Intersecting Families of Perfect Matchings

Abstract

A family of perfect matchings of $K_{2n}$ is $intersecting$ if any two of its members have an edge in common. It is known that if $\mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, then $|\mathcal{F}|\le (2n-3)!!$ and if equality holds, then $\mathcal{F}={\mathcal{F}}_{ij}$ where ${\mathcal{F}}_{ij}$ is the family of all perfect matchings of $K_{2n}$ that contain some fixed edge $ij$. In this note, we show that the extremal families are stable, namely, that for any $\epsilon \in (0,1/\sqrt{e})$ and $n>n(\epsilon )$, any intersecting family of perfect matchings of size greater than $(1-1/\sqrt{e}+\epsilon )(2n-3)!!$ is contained in ${\mathcal{F}}_{ij}$ for some edge $ij$. The proof uses the Gelfand pair $(S_{2n},S_2\wr S_n)$ along with an isoperimetric method of Ellis.