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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.18426 |
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| _version_ | 1866929607419101184 |
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| author | Jia, Zhen Xiang, Qing Xiao, Jimeng Zhang, Huajun |
| author_facet | Jia, Zhen Xiang, Qing Xiao, Jimeng Zhang, Huajun |
| contents | Let $m\geq 2$, $n$ be positive integers, and $R_i=\{k_{i,1} >k_{i,2} >\cdots> k_{i,t_i}\}$ be subsets of $[n]$ for $i=1,2,\ldots,m$. The families $\mathcal{F}_1\subseteq \binom{[n]}{R_1},\mathcal{F}_2\subseteq \binom{[n]}{R_2},\ldots,\mathcal{F}_m\subseteq \binom{[n]}{R_m}$ are said to be non-empty cross-intersecting if for each $i\in [m]$, $\mathcal{F}_i\neq\emptyset$ and for any $A\in \mathcal{F}_i,B\in\mathcal{F}_j$, $1\leq i<j\leq m$, $|A\bigcap B|\geq1$. In this paper, we determine the maximum value of $\sum_{j=1}^{m}|\mathcal{F}_j|$ for non-empty cross-intersecting family $\mathcal{F}_1, \mathcal{F}_2,\ldots,\mathcal{F}_m$ when $n\geq k_1+k_2$, where $k_1$ (respectively, $k_2$) is the largest (respectively, second largest) value in $\{k_{1,1},k_{2,1},\ldots,k_{m,1}\}$. This result is a generalization of the results by Shi, Frankl and Qian \cite{shi2022non} on non-empty cross-intersecting families. Moreover, the extremal families are completely characterized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18426 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-uniform Cross-intersecting Families Jia, Zhen Xiang, Qing Xiao, Jimeng Zhang, Huajun Combinatorics 05D05 Let $m\geq 2$, $n$ be positive integers, and $R_i=\{k_{i,1} >k_{i,2} >\cdots> k_{i,t_i}\}$ be subsets of $[n]$ for $i=1,2,\ldots,m$. The families $\mathcal{F}_1\subseteq \binom{[n]}{R_1},\mathcal{F}_2\subseteq \binom{[n]}{R_2},\ldots,\mathcal{F}_m\subseteq \binom{[n]}{R_m}$ are said to be non-empty cross-intersecting if for each $i\in [m]$, $\mathcal{F}_i\neq\emptyset$ and for any $A\in \mathcal{F}_i,B\in\mathcal{F}_j$, $1\leq i<j\leq m$, $|A\bigcap B|\geq1$. In this paper, we determine the maximum value of $\sum_{j=1}^{m}|\mathcal{F}_j|$ for non-empty cross-intersecting family $\mathcal{F}_1, \mathcal{F}_2,\ldots,\mathcal{F}_m$ when $n\geq k_1+k_2$, where $k_1$ (respectively, $k_2$) is the largest (respectively, second largest) value in $\{k_{1,1},k_{2,1},\ldots,k_{m,1}\}$. This result is a generalization of the results by Shi, Frankl and Qian \cite{shi2022non} on non-empty cross-intersecting families. Moreover, the extremal families are completely characterized. |
| title | Non-uniform Cross-intersecting Families |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2411.18426 |