An Erd\H{o}s--Ko--Rado theorem for matchings in the complete graph

TL;DR

This paper extends the Erd ext{"o}s--Ko--Rado theorem to matchings in complete graphs, proving maximum size of intersecting families and characterizing extremal cases using a cycle method.

Contribution

It establishes a higher-order Erd ext{"o}s--Ko--Rado theorem for matchings in complete graphs, identifying maximum intersecting families and their structure.

Findings

Maximum intersecting family size equals that of a star

Equality holds only for star families

Uses an analog of Katona's cycle method

Abstract

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.