  Two perfect matchings $P$ and $Q$ of the complete graph on $2k$ vertices are
said to be set-wise $t$-intersecting if there exist edges $P_{1}, \cdots,
P_{t}$ in $P$ and $Q_{1}, \cdots, Q_{t}$ in $Q$ such that the union of edges
$P_{1}, \cdots, P_{t}$ has the same set of vertices as the union of $Q_{1},
\cdots, Q_{t}$ has. In this paper we prove an extension of the famous
Erd\H{o}s-Ko-Rado (EKR) theorem to set-wise $2$-intersecting families of
perfect matching on all values of $k$, and we conjecture similar statement for
all $t\geq 2$.
