An Extension of the Erd\H{o}s-Ko-Rado Theorem to uniform set partitions

TL;DR

This paper extends the Erdős-Ko-Rado theorem to uniform set partitions, identifying the largest families of partitions with a fixed intersecting block property, especially for large parameters and specific cases.

Contribution

It proves a new version of the Erdős-Ko-Rado theorem for partially 2-intersecting uniform set partitions, including specific results for all  cases.

Findings

Largest 2-partially intersecting families contain a fixed pair in a block

Results hold for sufficiently large  and for all =3 cases

Extends classical intersecting set theory to set partitions

Abstract

A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left| P_{i} \cap Q_{j} \right|\geq t$. In this paper we prove a version of the Erd\H{o}s-Ko-Rado theorem for partially $2$-intersecting $(k,\ell)$-partitions. In particular, we show for $\ell$ sufficiently large, the set of all $(k,\ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,\ell)$-partitions. For for $k=3$, we show this result holds for all $\ell$.