Stability for 1-intersecting families of perfect matchings

Abstract

A family of perfect matchings of K2n is intersecting if any two of its members have an edge in common. It is known that if F is a family of intersecting perfect matchings of K2n, then |F|⩽(2n−3)!! and if equality holds, then F=Fij where Fij is the family of all perfect matchings of K2n that contain some fixed edge ij. In this work, we show that the extremal families are stable, namely, that for any ϵ∈(0,1∕e) and n>n(ϵ), any intersecting family of perfect matchings of size greater than (1−1∕e+ϵ)(2n−3)!! is contained in Fij for some edge ij. The proof uses the Gelfand pair (S2n,S2≀Sn) along with an isoperimetric method of Ellis.