An Analogue of Hilton-Milner Theorem for Set Partitions

TL;DR

This paper establishes an upper bound on the size of non-trivial t-intersecting families of set partitions, extending the Hilton-Milner theorem to the context of set partitions with a characterization of extremal families.

Contribution

It provides the first Hilton-Milner type theorem for set partitions, including an explicit bound and characterization of extremal families for large n.

Findings

Derived an upper bound for non-trivial t-intersecting set partition families.

Characterized the structure of extremal families achieving equality.

Extended classical intersection theorems to the setting of set partitions.

Abstract

Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal{A} \subseteq \mathcal{B}(n)$ is a non-trivial $t$-intersecting family of set partitions i.e. any two members of $\A$ have at least $t$ blocks in common, but there is no fixed $t$ blocks of size one which belong to all of them. It is proved that for sufficiently large $n$ depending on $t$, \[ |\mathcal{A}| \le B_{n-t}-\tilde{B}_{n-t}-\tilde{B}_{n-t-1}+t \] where $B_{n}$ is the $n$-th Bell number and $\tilde{B}_{n}$ is the number of set partitions of $[n]$ without blocks of size one. Moreover, equality holds if and only if $\mathcal{A}$ is equivalent to \[ \{P \in \mathcal{B}(n): \{1\}, \{2\},..., \{t\}, \{i\} \in P \textnormal{for some} i \not = 1,2,..., t,n \}\cup \{Q(i,n)\ :\ 1\leq i\leq t\} \] where $Q(i,n)=\{\{i,n\}\}\cup\{\{j\}\ :\ j\in [n]\setminus \{i,n\}\}$. This is an analogue of the Hilton-Milner theorem for set partitions.