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.
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