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Consider the family of all perfect matchings of the complete graph $K_{2n}$with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings ofsize $s$, 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 giveseveral lower bounds for $f(n,x)$. We also apply the Lov\'asz local lemma tofind a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ timesthen 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 resultvis-\'a-vis the covering radius of sets of permutations, which was studied byCameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}).We also conclude with a conjecture of a more general problem in hypergraphmatchings.

The covering radius problem for sets of perfect matchings