## An Erdős-Ko-Rado theorem for subset partitions.(English)Zbl 1309.05176

Summary: A $$k\ell$$-subset partition, or $$(k,\ell)$$-subpartition, is a $$k\ell$$-subset of an $$n$$-set that is partitioned into $$\ell$$ distinct blocks, each of size $$k$$. Two $$(k,\ell)$$-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 $$(k,\ell)$$-subpartitions. We show that for $$n \geq k\ell$$, $$\ell \geq 2$$ and $$k \geq 3$$, the number of $$(k,\ell)$$-subpartitions in the largest 1-intersecting family is at most $$\binom{n-k}{k}\binom{n-2k}{k}\cdots\binom{n-(\ell-1)k}{k}/(\ell-1)!$$, and that this bound is only attained by the family of $$(k,\ell)$$-subpartitions with a common fixed block, known as the canonical intersecting family of $$(k,\ell)$$-subpartitions. Further, provided that $$n$$ is sufficiently large relative to $$k,\ell$$ and $$t$$, the largest $$t$$-intersecting family is the family of $$(k,\ell)$$-subpartitions that contain a common set of $$t$$ fixed blocks.

### MSC:

 05D05 Extremal set theory 05A18 Partitions of sets