Leírás
A perfect matching of a complete graph $K_{2n}$ is a 1-regular subgraph that contains all the vertices.
Two perfect matchings intersect if they share an edge. It is known that if $\mathcal{F}$ is family of
intersecting perfect matchings of $K_{2n}$, then $|\mathcal{F}| \leq (2(n-1) - 1)!!$ 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 $ij$. We give a short algebraic proof of this result, resolving a
question of Godsil and Meagher. Along the way, we show that if a family $\mathcal{F}$ is non-Hamiltonian,
that is, $m \cup m' \not \cong C_{2n}$ for any $m,m' \in \mathcal{F}$, then $|\mathcal{F}| \leq (2(n-1) - 1)!!$
and this bound is met with equality if and only if $\mathcal{F} = \mathcal{F}_{ij}$. Our results make
ample use of a somewhat understudied symmetric commutative association scheme arising from the
Gelfand pair $(S_{2n},S_2 \wr S_n)$. We give an exposition of a few new interesting objects that live
in this scheme as they pertain to our results.