An Erdős–Ko–Rado theorem for subset partitions

Abstract

A kℓ-subset partition, or (k, ℓ)-subpartition, is a kℓ-subset of an n-set that is partitioned into ℓ distinct classes, each of size k. Two (k, ℓ)-subpartitions are said to t-intersect if they have at least t classes in common. In this paper, we prove an Erdős-Ko-Rado theorem for intersecting families of (k, ℓ)-subpartitions. We show that for n ≥ kℓ, ℓ ≥ 2 and k ≥ 3, the largest 1-intersecting family contains at most-subpartitions, and that this bound is only attained by the family of (k, ℓ)-subpartitions with a common fixed class, known as the canonical intersecting family of (k, ℓ)-subpartitions. Further, provided that n is sufficiently large relative to k, ℓ and t, the largest t-intersecting family is the family of (k, ℓ)-subpartitions that contain a common set of t fixed classes.