On $r$-cross $t$-intersecting families of partitions

Abstract

In this paper, we address several intersection problems for $r$-cross $t$-intersecting families of partitions. A $k$-partition of an $n$-set $X$ is a set of $k$ pairwise disjoint non-empty subsets whose union is $X$. For $1\le i\le r$, let ${\mathcal{F}}_i$ be a family of $k_i$-partitions of $X$. We say that ${\mathcal{F}}_1,{\mathcal{F}}_2,\dots ,{\mathcal{F}}_r$ are $r$-cross $t$-intersecting if $|{\cap }_{i=1}^r{F}_i|\ge t$ for all $F_i\in {\mathcal{F}}_i$. The families are called non-trivial if $|{\cap }_{i=1}^r({\cap }_{F\in {\mathcal{F}}_i}F)|<t$. Proving an Erdős-Ko-Rado type theorem, we determine the families maximizing ${\prod }_{i=1}^r|{\mathcal{F}}_i|$. We further determine non-trivial $r$-cross $t$-intersecting families with maximum product of sizes; this result also serves as a Hilton-Milner type theorem. In particular, for $r=2$ there are two potential structures for optimal families, and for $r\ge 3$ exactly one remains.