Hilton-Milner theorem for k-multisets

2023/08/07 by Liao, Jiaqi, Lv, Zequn, Cao, Mengyu +1 · 1 citation
#05A15 #05C35 #05D05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.03585

Abstract

Let k, n ∈ ℕ⁺ and m ∈ ℕ⁺ ∪ \∞ \ . A k -multiset in [n]_m is a k -set whose elements are integers from \1, 2, …, 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 ≥ k + \lceil k/m \rceil . In the special case when m=∞, 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.

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• Hilton-Milner Theorem for the r-independent sets in a union of cliques

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