## Involve: A Journal of Mathematics

- Involve
- Volume 8, Number 1 (2015), 119-127.

### 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 $\left(\genfrac{}{}{0.0pt}{}{n-k}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n-2k}{k}\right)\cdots \left(\genfrac{}{}{0.0pt}{}{n-\left(\ell -1\right)k}{k}\right)\u2215\left(\ell -1\right)!$, 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.

#### Article information

**Source**

Involve, Volume 8, Number 1 (2015), 119-127.

**Dates**

Received: 3 October 2013

Revised: 9 April 2014

Accepted: 12 April 2014

First available in Project Euclid: 22 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1511370844

**Digital Object Identifier**

doi:10.2140/involve.2015.8.119

**Mathematical Reviews number (MathSciNet)**

MR3321715

**Zentralblatt MATH identifier**

1309.05176

**Subjects**

Primary: 05D05: Extremal set theory

**Keywords**

Erdős–Ko–Rado theorem set partitions

#### Citation

Dyck, Adam; Meagher, Karen. An Erdős–Ko–Rado theorem for subset partitions. Involve 8 (2015), no. 1, 119--127. doi:10.2140/involve.2015.8.119. https://projecteuclid.org/euclid.involve/1511370844