Hilton-Milner theorem for $k$-multisets

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.