On cross-intersecting families of set partitions
Summary: 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 \(n\), \(|{\mathcal{A}}_1| |{\mathcal{A}}_2| \leq B_{n-1}^2 \) 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.
scientific article; zbMATH DE number 6171832