On $r$-cross $t$-intersecting families of partitions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908848474816512 |
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| author | Wen, Jie Lv, Benjian |
| author_facet | Wen, Jie Lv, Benjian |
| contents | In this paper, we address several intersection problems for $r$-cross $t$-intersecting families of partitions. A $k$-partition of an $n$-set $X$ is a set of $k$ pairwise disjoint non-empty subsets whose union is $X$. For $1\leq i\leq r$, let $\mathcal{F}_i$ be a family of $k_i$-partitions of $X$. We say that $\mathcal{F}_1,\mathcal{F}_2,\ldots,\mathcal{F}_r$ are $r$-cross $t$-intersecting if $|\cap_{i=1}^{r}F_i|\geq t$ for all $F_i\in\mathcal{F}_i$. The families are called non-trivial if $|\cap_{i=1}^r(\cap_{F\in\mathcal{F}_i}F)|<t$. Proving an Erdős-Ko-Rado type theorem, we determine the families maximizing $\prod_{i=1}^r|\mathcal{F}_i|$. We further determine non-trivial $r$-cross $t$-intersecting families with maximum product of sizes; this result also serves as a Hilton-Milner type theorem. In particular, for $r=2$ there are two potential structures for optimal families, and for $r\geq3$ exactly one remains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19464 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $r$-cross $t$-intersecting families of partitions Wen, Jie Lv, Benjian Combinatorics 05D05 In this paper, we address several intersection problems for $r$-cross $t$-intersecting families of partitions. A $k$-partition of an $n$-set $X$ is a set of $k$ pairwise disjoint non-empty subsets whose union is $X$. For $1\leq i\leq r$, let $\mathcal{F}_i$ be a family of $k_i$-partitions of $X$. We say that $\mathcal{F}_1,\mathcal{F}_2,\ldots,\mathcal{F}_r$ are $r$-cross $t$-intersecting if $|\cap_{i=1}^{r}F_i|\geq t$ for all $F_i\in\mathcal{F}_i$. The families are called non-trivial if $|\cap_{i=1}^r(\cap_{F\in\mathcal{F}_i}F)|<t$. Proving an Erdős-Ko-Rado type theorem, we determine the families maximizing $\prod_{i=1}^r|\mathcal{F}_i|$. We further determine non-trivial $r$-cross $t$-intersecting families with maximum product of sizes; this result also serves as a Hilton-Milner type theorem. In particular, for $r=2$ there are two potential structures for optimal families, and for $r\geq3$ exactly one remains. |
| title | On $r$-cross $t$-intersecting families of partitions |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2602.19464 |