Consider the family of all perfect matchings of the complete graph $K_{2n}$
with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of
size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$,
then there exists a perfect matching that agrees with each perfect matching in
$\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give
several lower bounds for $f(n,x)$. We also apply the Lov\'asz local lemma to
find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times
then there exists a perfect matching that agrees with each perfect matching in
$\mathcal M$ in at most $x-1$ edges. This is an analogue of an extremal result
vis-\'a-vis the covering radius of sets of permutations, which was studied by
Cameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}).
We also conclude with a conjecture of a more general problem in hypergraph
matchings.