The Erdős-Ko-Rado theorem for \(2\)-intersecting families of perfect matchings

A perfect matching in the complete graph on \(2k\) vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exactly once. Two perfect matchings are said to be \(t\)-intersecting if they have at least \(t\) edges in common. The main result in this paper is an extension of the famous Erdős-Ko-Rado (EKR) theorem \cite{EKR} to 2-intersecting families of perfect matchings for all values of \(k\). Specifically, for \(k\geq 3\) a set of 2-intersecting perfect matchings in \(K_{2k}\) of maximum size has \((2k-5)(2k-7)\cdots (1)\) perfect matchings.