On extremal cross $t$-intersecting families with $t$-covering number conditions
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| Format: | Preprint |
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2026
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| _version_ | 1866914579504693248 |
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| author | Zhu, Yu Lv, Benjian Wang, Kaishun |
| author_facet | Zhu, Yu Lv, Benjian Wang, Kaishun |
| contents | Let $n$, $k$ and $t$ be positive integers, and let $\mathcal{F}$ be a collection of $k$-subsets of $[n]=\{1,2,\dots,n\}$. The $t$-covering number $τ_t(\mathcal{F})$ of $\mathcal{F}$ is defined as the minimum size of a set $T$ such that $|F\cap T|\geq t$ for all $F\in \mathcal{F}$. For positive integers $k_1$ and $k_2$, let $\mathcal{F}_i$ be a collection of $k_i$-subsets of $[n]$ for $i\in \{1,2\}$. The families $\mathcal{F}_1$ and $\mathcal{F}_2$ are said to be cross $t$-intersecting if $|F_1\cap F_2|\geq t$ for all $F_1\in\mathcal{F}_1$ and $F_2\in \mathcal{F}_2$. When $\mathcal{F}_1=\mathcal{F}_2$, $\mathcal{F}_1$ is called a $t$-intersecting family. In this paper, we first characterize the extremal structures of cross $t$-intersecting families $\mathcal{F}_1$ and $\mathcal{F}_2$ that maximize $|\mathcal{F}_1||\mathcal{F}_2|$ under the condition that $τ_t(\mathcal{F}_1)\geq t+1$ and $τ_t(\mathcal{F}_2)\geq t+1$. We then describe the maximal $t$-intersecting families with $t$-covering number $t+1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_19424 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On extremal cross $t$-intersecting families with $t$-covering number conditions Zhu, Yu Lv, Benjian Wang, Kaishun Combinatorics 05D05, 05A10 Let $n$, $k$ and $t$ be positive integers, and let $\mathcal{F}$ be a collection of $k$-subsets of $[n]=\{1,2,\dots,n\}$. The $t$-covering number $τ_t(\mathcal{F})$ of $\mathcal{F}$ is defined as the minimum size of a set $T$ such that $|F\cap T|\geq t$ for all $F\in \mathcal{F}$. For positive integers $k_1$ and $k_2$, let $\mathcal{F}_i$ be a collection of $k_i$-subsets of $[n]$ for $i\in \{1,2\}$. The families $\mathcal{F}_1$ and $\mathcal{F}_2$ are said to be cross $t$-intersecting if $|F_1\cap F_2|\geq t$ for all $F_1\in\mathcal{F}_1$ and $F_2\in \mathcal{F}_2$. When $\mathcal{F}_1=\mathcal{F}_2$, $\mathcal{F}_1$ is called a $t$-intersecting family. In this paper, we first characterize the extremal structures of cross $t$-intersecting families $\mathcal{F}_1$ and $\mathcal{F}_2$ that maximize $|\mathcal{F}_1||\mathcal{F}_2|$ under the condition that $τ_t(\mathcal{F}_1)\geq t+1$ and $τ_t(\mathcal{F}_2)\geq t+1$. We then describe the maximal $t$-intersecting families with $t$-covering number $t+1$. |
| title | On extremal cross $t$-intersecting families with $t$-covering number conditions |
| topic | Combinatorics 05D05, 05A10 |
| url | https://arxiv.org/abs/2605.19424 |