On Cross-Intersecting Families of Set Partitions

Abstract

Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal A_1,\mathcal A_2\subseteq \mathcal{B}(n)$ are cross-intersecting i.e. for all $A_1\in \mathcal A_1$ and $A_2\in \mathcal A_2$, we have $A_1\cap A_2\neq\varnothing$. It is proved that for sufficiently large $n$,
\[ \vert \mathcal A_1\vert\vert \mathcal A_2\vert\leq B_{n-1}^2
\]
where $B_{n}$ is the $n$-th Bell number. Moreover, equality holds if and only if $\mathcal{A}_1=\mathcal A_2$ and $\mathcal A_1$ consists of all set partitions with a fixed singleton.