  We consider the following higher-order analog of the Erd\H{o}s--Ko--Rado
theorem. For positive integers r and n with r<= n, let M^r_n be the family of
all matchings of size r in the complete graph K_{2n}. For any edge e in
E(K_{2n}), the family M^r_n(e), which consists of all sets in M^r_n containing
e, is called the star centered at e. We prove that if r<n and A is an
intersecting family of matchings in M^r_n, then |A|<=|M^r_n(e)|$, where e is an
edge in E(K_{2n}). We also prove that equality holds if and only if A is a
star. The main technique we use to prove the theorem is an analog of Katona's
elegant cycle method.
