An analogue of the Erd{\H o}s Matching Conjecture for permutations with fixed number of cycles

Abstract

Let $S_n$ denote the set of permutations of $[n]=\{1,2,\dots ,n\}$. For each integer $k\ge 1$, let $S_{n,k}$ be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles. A subset $H\subseteq S_{n,k}$ is to be a matching if ${\pi }_1$ and ${\pi }_2$ do not have any common cycles for all distinct ${\pi }_1,{\pi }_2\in H$. The matching number of a family $\mathcal{A}\subseteq S_{n,k}$ is denoted by ${\nu }_p(\mathcal{A})$ and is defined to be the size of the largest matching in $\mathcal{A}$. In this paper, we determine the maximum size of a family $\mathcal{A}\subseteq S_{n,k}$ subject to the condition ${\nu }_p(\mathcal{A})\le s$.