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

TL;DR

This paper investigates the maximum product sizes of $r$-cross $t$-intersecting families of partitions, extending classical intersection theorems to partition families and identifying optimal structures.

Contribution

It establishes Erd ext{o}s-Ko-Rado and Hilton-Milner type theorems for $r$-cross $t$-intersecting families of partitions, including non-trivial cases and unique optimal structures.

Findings

Determined maximum product sizes for $r$-cross $t$-intersecting families.

Identified two potential structures for optimal families when $r=2$.

Proved unique optimal structure for $r extgreater 2$.

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 $1\leq i\leq r$, let $\mathcal{F}_i$ be a family of $k_i$-partitions of $X$. We say that $\mathcal{F}_1,\mathcal{F}_2,\ldots,\mathcal{F}_r$ are $r$-cross $t$-intersecting if $|\cap_{i=1}^{r}F_i|\geq t$ 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 Erd\H{o}s-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 $r\geq3$ exactly one remains.