Hilton-Milner theorem for $k$-multisets

TL;DR

This paper extends the Hilton-Milner theorem to $k$-multisets, providing the maximum size and structure of largest non-trivial intersecting families in multisets with bounded or unbounded repetitions.

Contribution

It generalizes the Hilton-Milner theorem to multisets, unifying finite set and unbounded multiset cases, and characterizes the structure of maximal intersecting families.

Findings

Largest non-trivial intersecting family size determined for $k$-multisets.

Unifies finite set and unbounded multiset Hilton-Milner results.

Provides structural description of extremal families.

Abstract

Let $ k, n \in \mathbb{N}^+ $ and $ m \in \mathbb{N}^+ \cup \{\infty \} $. A $ k $-multiset in $ [n]_m $ is a $ k $-set whose elements are integers from $ \{1, 2, \ldots, n\} $, and each element is allowed to have at most $ m $ repetitions. A family of $ k $-multisets in $ [n]_m $ is said to be intersecting if every pair of $ k $-multisets from the family have non-empty intersection. In this paper, we give the size and structure of the largest non-trivial intersecting family of $ k $-multisets in $ [n]_m $ for $ n \geq k + \lceil k/m \rceil $. In the special case when $m=\infty$, our result gives rise to an unbounded multiset version for Hilton-Milner Theorem given by Meagher and Purdy. Furthermore, our main theorem unites the statements of the Hilton-Milner Theorem for finite sets and unbounded multisets.