Erdős-Ko-Rado theorem and Hilton-Milner type theorem for $k$-partitions
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2025
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| _version_ | 1866908608867860480 |
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| author | Wen, Jie Lv, Benjian |
| author_facet | Wen, Jie Lv, Benjian |
| contents | A $k$-partition of an $n$-set $X$ is a collection of $k$ pairwise disjoint non-empty subsets whose union is $X$. A family of $k$-partitions of $X$ is called $t$-intersecting if any two of its members share at least $t$ blocks. A $t$-intersecting family is trivial if every $k$-partition in it contains $t$ fixed blocks, and is non-trivial otherwise. In this paper, we first prove that, for $n\geq L(k,t):=(t+1)+(k-t+1)\cdot\log_2(t+1)(k-t+1)$, a $t$-intersecting family with maximum size must consist of all $k$-partitions containing $t$ fixed singletons. This improves the results given by Erdős and Székely (2000), and by Kupavskii (2023). We further determine the non-trivial $t$-intersecting families of $k$-partitions with maximum size for $n \ge 2L(k,t)$, which turn out to be natural analogs of the corresponding families for finite sets. In addition, we prove a stability result. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_20251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Erdős-Ko-Rado theorem and Hilton-Milner type theorem for $k$-partitions Wen, Jie Lv, Benjian Combinatorics 05D05 A $k$-partition of an $n$-set $X$ is a collection of $k$ pairwise disjoint non-empty subsets whose union is $X$. A family of $k$-partitions of $X$ is called $t$-intersecting if any two of its members share at least $t$ blocks. A $t$-intersecting family is trivial if every $k$-partition in it contains $t$ fixed blocks, and is non-trivial otherwise. In this paper, we first prove that, for $n\geq L(k,t):=(t+1)+(k-t+1)\cdot\log_2(t+1)(k-t+1)$, a $t$-intersecting family with maximum size must consist of all $k$-partitions containing $t$ fixed singletons. This improves the results given by Erdős and Székely (2000), and by Kupavskii (2023). We further determine the non-trivial $t$-intersecting families of $k$-partitions with maximum size for $n \ge 2L(k,t)$, which turn out to be natural analogs of the corresponding families for finite sets. In addition, we prove a stability result. |
| title | Erdős-Ko-Rado theorem and Hilton-Milner type theorem for $k$-partitions |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2510.20251 |