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The covering radius problem for sets of perfect matchings
Consider the family of all perfect matchings of the complete graph
with vertices. Given any collection of perfect matchings of
size , there exists a maximum number such that if ,
then there exists a perfect matching that agrees with each perfect matching in
in at most edges. We use probabilistic arguments to give
several lower bounds for . We also apply the Lov\'asz local lemma to
find a function such that if each edge appears at most times
then there exists a perfect matching that agrees with each perfect matching in
in at most 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.Comment: 10 page
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