Intersection theorems for multisets

TL;DR

This paper extends classical intersection theorems to multisets, establishing new bounds and structures for intersecting and t-intersecting families of multisets using graph homomorphisms.

Contribution

It introduces multiset versions of the Hilton-Milner theorem and characterizes largest t-intersecting multiset families under certain conditions.

Findings

Established a multiset version of the Hilton-Milner theorem.

Characterized the maximum size and structure of t-intersecting multiset families when m ≤ 2k - t.

Applied graph homomorphisms to extend classical set system results to multisets.

Abstract

Let $k$, $t$ and $m$ be positive integers. A $k$-multiset of $[m]$ is a collection of $k$ integers from the set $\{1,...,m\}$ in which the integers can appear more than once. We use graph homomorphisms and existing theorems for intersecting and $t$-intersecting $k$-set systems to prove new results for intersecting and $t$-intersecting families of $k$-multisets. These results include a multiset version of the Hilton-Milner theorem and a theorem giving the size and structure of the largest $t$-intersecting family of $k$-multisets of an $m$-set when $m \leq 2k-t$.