An Erdős–Ko–Rado theorem for partial permutations

Abstract

Let $[n]$ denote the set of positive integers $\{1,2,\dots ,n\}$. An r-partial permutation of $[n]$ is a pair $(A,f)$ where $A\subseteq [n]$, $\left|A\right|=r$ and $f:A\to [n]$ is an injective map. A set $\mathcal{A}$ of r-partial permutations is intersecting if for any $(A,f)$, $(B,g)\in \mathcal{A}$, there exists $x\in A\cap B$ such that $f(x)=g(x)$. We prove that for any intersecting family $\mathcal{A}$ of r-partial permutations, we have $\left|\mathcal{A}\right|⩽\left(\genfrac{}{}{0em}{}{n-1}{r-1}\right)((n-1)!/(n-r)!)$.

It seems rather hard to characterize the case of equality. For $8⩽r⩽n-3$, we show that equality holds if and only if there exist $x_0$ and ${\epsilon }_0$ such that $\mathcal{A}$ consists of all $(A,f)$ for which $x_0\in A$ and $f(x_0)={\epsilon }_0$.