Almost covers of finite sets of points
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910103608754176 |
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| author | Hegedüs, Gábor |
| author_facet | Hegedüs, Gábor |
| contents | Let $\mbox{$\cal V$} \subseteq {\mathbb F}^n$ be a finite set of points in an affine space. A finite set of affine hyperplanes $\{H_1, \ldots ,H_m\}$ is said to be an almost cover of $\mbox{$\cal V$}$ and $\mathbf{v}$, if their union $\cup_{j=1}^m H_j$ contains $\mbox{$\cal V$}\setminus \{\mathbf{v}\}$ but does not contain $\mathbf{v}$. We give here a lower bound for the size of a minimal almost cover of $\mbox{$\cal V$}$ and $\mathbf{v}$ in terms of the size of $\mbox{$\cal V$}$ and the dimension $n$. We prove a generalization of Sziklai and Weiner's Theorem. Our simple proof is based on Gröbner basis theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16231 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost covers of finite sets of points Hegedüs, Gábor Combinatorics 52C35, 13P10, 05D40 Let $\mbox{$\cal V$} \subseteq {\mathbb F}^n$ be a finite set of points in an affine space. A finite set of affine hyperplanes $\{H_1, \ldots ,H_m\}$ is said to be an almost cover of $\mbox{$\cal V$}$ and $\mathbf{v}$, if their union $\cup_{j=1}^m H_j$ contains $\mbox{$\cal V$}\setminus \{\mathbf{v}\}$ but does not contain $\mathbf{v}$. We give here a lower bound for the size of a minimal almost cover of $\mbox{$\cal V$}$ and $\mathbf{v}$ in terms of the size of $\mbox{$\cal V$}$ and the dimension $n$. We prove a generalization of Sziklai and Weiner's Theorem. Our simple proof is based on Gröbner basis theory. |
| title | Almost covers of finite sets of points |
| topic | Combinatorics 52C35, 13P10, 05D40 |
| url | https://arxiv.org/abs/2405.16231 |