An Erdős–Ko–Rado theorem for subset partitions

Abstract

A $k\ell$-subset partition, or $\left(k,\ell \right)$-subpartition, is a $k\ell$-subset of an $n$-set that is partitioned into $\ell$ distinct blocks, each of size $k$. Two $\left(k,\ell \right)$-subpartitions are said to $t$-intersect if they have at least $t$ blocks in common. In this paper, we prove an Erdős–Ko–Rado theorem for intersecting families of $\left(k,\ell \right)$-subpartitions. We show that for $n\ge k\ell$, $\ell \ge 2$ and $k\ge 3$, the number of $\left(k,\ell \right)$-subpartitions in the largest $1$-intersecting family is at most $\left(\genfrac{}{}{0.0pt}{}{n-k}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n-2k}{k}\right)\cdots \left(\genfrac{}{}{0.0pt}{}{n-\left(\ell -1\right)k}{k}\right)∕\left(\ell -1\right)!$, and that this bound is only attained by the family of $\left(k,\ell \right)$-subpartitions with a common fixed block, known as the canonical intersecting family of $\left(k,\ell \right)$-subpartitions. Further, provided that $n$ is sufficiently large relative to $k,\ell$ and $t$, the largest $t$-intersecting family is the family of $\left(k,\ell \right)$-subpartitions that contain a common set of $t$ fixed blocks.