Stability for Intersecting Families of Perfect Matchings

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 $|\mathcal{F}| \leq (2n-3)!!$ 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.