An Erdős–Ko–Rado Theorem for Matchings

The Erdős–Ko–Rado theorem says that if r ≤ n/2, any family of r-subsets of an n-element set in which every two members share at least one element is no bigger than a star (all r-subsets containing one fixed element). I ask the same question for matchings with r edges in K_2n, where two matchings intersect if they have an edge in common. For r < n, an intersecting family is never larger than the star of all r-matchings through a fixed edge, and only stars reach that bound. The proof adapts Katona's cycle method, building its cyclic orders from partitions of the edges of K_2n into perfect matchings.