Involve: A Journal of Mathematics

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

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 blocks, each of size $k$. Two $(k,ℓ)$-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 $(k,ℓ)$-subpartitions. We show that for $n ≥ kℓ$, $ℓ ≥ 2$ and $k ≥ 3$, the number of $(k,ℓ)$-subpartitions in the largest $1$-intersecting family is at most $n−k k n−2k k ⋯n−(ℓ−1)k k ∕(ℓ − 1)!$, and that this bound is only attained by the family of $(k,ℓ)$-subpartitions with a common fixed block, 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 blocks.

Article information

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

Dates
Revised: 9 April 2014
Accepted: 12 April 2014
First available in Project Euclid: 22 November 2017

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

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

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