An Erd\H{o}s-Ko-Rado theorem for multisets

TL;DR

This paper extends the Erd ext{o}s-Ko-Rado theorem to multisets, determining the maximum size and structure of intersecting collections of multisets for various parameter ranges.

Contribution

It establishes the maximum size and structure of intersecting multiset collections, generalizing the classical Erd ext{o}s-Ko-Rado theorem to multisets.

Findings

Maximum size of intersecting multiset collections for m ≥ k+1

Characterization of extremal collections when m > k+1

Results for the case m ≤ k

Abstract

Let $k$ and $m$ be positive integers. A collection of $k$-multisets from $\{1,..., m \}$ is intersecting if every pair of multisets from the collection is intersecting. We prove that for $m \geq k+1$, the size of the largest such collection is $\binom{m+k-2}{k-1}$ and that when $m > k+1$, only a collection of all the $k$-multisets containing a fixed element will attain this bound. The size and structure of the largest intersecting collection of $k$-multisets for $m \leq k$ is also given.