Intersecting generalised permutations

Abstract

For any positive integers $k,r,n$ with $r \leq \min\{k,n\}$, let $\mathcal{P}_{k,r,n}$ be the family of all sets $\{(x_1,y_1), \dots, (x_r,y_r)\}$ such that $x_1, \dots, x_r$ are distinct elements of $[k] = \{1, \dots, k\}$ and $y_1, \dots, y_r$ are distinct elements of $[n]$. The families $\mathcal{P}_{n,n,n}$ and $\mathcal{P}_{n,r,n}$ describe permutations of $[n]$ and $r$-partial permutations of $[n]$, respectively. If $k \leq n$, then $\mathcal{P}_{k,k,n}$ describes permutations of $k$-element subsets of $[n]$. A family $\mathcal{A}$ of sets is said to be intersecting if every two members of $\mathcal{A}$ intersect. In this note we use Katona's elegant cycle method to show that a number of important Erd\H{o}s-Ko-Rado-type results by various authors generalise as follows: the size of any intersecting subfamily $\mathcal{A}$ of $\mathcal{P}_{k,r,n}$ is at most ${k-1 \choose r-1}\frac{(n-1)!}{(n-r)!}$, and the bound is attained if and only if $\mathcal{A} = \{A \in \mathcal{P}_{k,r,n} \colon (a,b) \in A\}$ for some $a \in [k]$ and $b \in [n]$.