  A family of perfect matchings of $K_{2n}$ is $t$-$intersecting$ if any two
members share $t$ or more edges. We prove for any $t \in \mathbb{N}$ that every
$t$-intersecting family of perfect matchings has size no greater than $(2(n-t)
- 1)!!$ for sufficiently large $n$, and that equality holds if and only if the
family is composed of all perfect matchings that contain a fixed set of $t$
disjoint edges. This is an asymptotic version of a conjecture of Godsil and
Meagher that can be seen as the non-bipartite analogue of the Deza-Frankl
conjecture proven by Ellis, Friedgut, and Pilpel.
