Hilton-Milner theorem for $k$-multisets

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 $\{1,2,\dots ,n\}$, 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 $n\ge k+\lceil k/m\rceil$. 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.

14 pages