The maximum sum of sizes of non-empty cross $t$-intersecting families
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914669568983040 |
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| author | Li, Shuang Liu, Dehai Song, Deping Yao, Tian |
| author_facet | Li, Shuang Liu, Dehai Song, Deping Yao, Tian |
| contents | Let $[n]:=\lbrace 1,2,\ldots,n \rbrace$, and $M$ be a set of positive integers. Denote the family of all subsets of $[n]$ with sizes in $M$ by $\binom{\left[n\right]}{M}$. The non-empty families $\mathcal{A}\subseteq\binom{\left[n\right]}{R}$ and $\mathcal{B}\subseteq \binom{\left[n\right]}{S}$ are said to be cross $t$-intersecting if $|A\cap B|\geq t$ for all $A\in \mathcal{A}$ and $B\in \mathcal{B}$. In this paper, we determine the maximum sum of sizes of non-empty cross $t$-intersecting families, and characterize the extremal families. Similar result for finite vector spaces is also proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00253 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The maximum sum of sizes of non-empty cross $t$-intersecting families Li, Shuang Liu, Dehai Song, Deping Yao, Tian Combinatorics Let $[n]:=\lbrace 1,2,\ldots,n \rbrace$, and $M$ be a set of positive integers. Denote the family of all subsets of $[n]$ with sizes in $M$ by $\binom{\left[n\right]}{M}$. The non-empty families $\mathcal{A}\subseteq\binom{\left[n\right]}{R}$ and $\mathcal{B}\subseteq \binom{\left[n\right]}{S}$ are said to be cross $t$-intersecting if $|A\cap B|\geq t$ for all $A\in \mathcal{A}$ and $B\in \mathcal{B}$. In this paper, we determine the maximum sum of sizes of non-empty cross $t$-intersecting families, and characterize the extremal families. Similar result for finite vector spaces is also proved. |
| title | The maximum sum of sizes of non-empty cross $t$-intersecting families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.00253 |