An extension of the Erd\H{o}s-Ko-Rado theorem to set-wise $2$-intersecting families of perfect matchings

TL;DR

This paper extends the Erd ext{"o}s-Ko-Rado theorem to set-wise 2-intersecting families of perfect matchings in complete graphs, providing new bounds and conjectures for larger t values.

Contribution

It proves an extension of the EKR theorem for set-wise 2-intersecting perfect matchings and proposes a conjecture for all t ≥ 2.

Findings

Extended EKR theorem to set-wise 2-intersecting perfect matchings

Established bounds for all k in this context

Formulated conjecture for t ≥ 2

Abstract

Two perfect matchings $P$ and $Q$ of the complete graph on $2k$ vertices are said to be set-wise $t$-intersecting if there exist edges $P_{1}, \cdots, P_{t}$ in $P$ and $Q_{1}, \cdots, Q_{t}$ in $Q$ such that the union of edges $P_{1}, \cdots, P_{t}$ has the same set of vertices as the union of $Q_{1}, \cdots, Q_{t}$ has. In this paper we prove an extension of the famous Erd\H{o}s-Ko-Rado (EKR) theorem to set-wise $2$-intersecting families of perfect matching on all values of $k$, and we conjecture similar statement for all $t\geq 2$.