An extension of the Erdős-Ko-Rado theorem to set-wise $2$-intersecting families of perfect matchings

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 Erdős-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 $t\ge 2$.

arXiv admin note: text overlap with arXiv:2008.08503