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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.17985 |
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| _version_ | 1866929606247841792 |
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| author | Shan, Yunjing Zhou, Junling |
| author_facet | Shan, Yunjing Zhou, Junling |
| contents | Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $\left[V\atop k\right]_q$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ is called intersecting if for all $F$, $F'\in\mathcal{F}$, we have ${\rm dim}$$(F\cap F')\geq 1$. Let $δ_{d}(\mathcal{F})$ denote the minimum degree in $\mathcal{F}$ of all $d$-dimensional subspaces. In this paper we show that $δ_{d}(\mathcal{F})\leq \left[n-d-1\atop k-d-1\right]$ in any intersecting family $\mathcal{F}\subseteq \left[V\atop k\right]_q$, where $k>d\geq 2$ and $n\geq 2k+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $d$-Degree Erdős-Ko-Rado theorem for finite vector spaces Shan, Yunjing Zhou, Junling Combinatorics Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $\left[V\atop k\right]_q$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ is called intersecting if for all $F$, $F'\in\mathcal{F}$, we have ${\rm dim}$$(F\cap F')\geq 1$. Let $δ_{d}(\mathcal{F})$ denote the minimum degree in $\mathcal{F}$ of all $d$-dimensional subspaces. In this paper we show that $δ_{d}(\mathcal{F})\leq \left[n-d-1\atop k-d-1\right]$ in any intersecting family $\mathcal{F}\subseteq \left[V\atop k\right]_q$, where $k>d\geq 2$ and $n\geq 2k+1$. |
| title | $d$-Degree Erdős-Ko-Rado theorem for finite vector spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.17985 |