The Erd\H{o}s-Ko-Rado theorem for $2$-intersecting families of perfect matchings

TL;DR

This paper extends the Erd ext{"o}s-Ko-Rado theorem to 2-intersecting families of perfect matchings in complete graphs, identifying the maximum size of such families for all $k \

Contribution

It generalizes the EKR theorem to 2-intersecting perfect matchings, providing exact maximum sizes for all $k \

Findings

Maximum size of 2-intersecting perfect matching families is $(2k-5)(2k-7) imes \

The result applies for all $k \\geq 3$.

The paper establishes a new combinatorial bound for intersecting matchings.

Abstract

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 Erd\H{o}s-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.