Hilton-Milner theorem for $k$-multisets
| Autor: | Liao, Jiaqi, Lv, Zequn, Cao, Mengyu, Lu, Mei |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 14 pages |
| Databáze: | arXiv |
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