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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.20772 |
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| _version_ | 1866915359508922368 |
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| author | Abrams, Aaron |
| author_facet | Abrams, Aaron |
| contents | Let $S \subseteq \mathbb{R}^n$, and let $k\in\mathbb{N}$. Greenwell and Johnson define ${\hatχ }^{(k)}(S)$ to be the smallest integer $m$ (if such an integer exists) such that for every $k\times m$ array $D=(d_{ij})$ of positive real numbers, $S$ can be colored with the colors $C_1,\ldots,C_m$ such that no two points of $S$ which are a (Euclidean) distance $d_{ij}$ apart are both colored $C_j$, for all $1\leq i \leq k$ and $1\leq j \leq m$. If no such integer exists then we say that ${\hatχ }^{(k)}(S)=\infty$. In this paper we show that ${\hatχ }^{(k)}(\mathbb{R})$ is finite for all $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $k^{\text th}$ Upper Chromatic Number of the Line Abrams, Aaron Combinatorics 05C15 Let $S \subseteq \mathbb{R}^n$, and let $k\in\mathbb{N}$. Greenwell and Johnson define ${\hatχ }^{(k)}(S)$ to be the smallest integer $m$ (if such an integer exists) such that for every $k\times m$ array $D=(d_{ij})$ of positive real numbers, $S$ can be colored with the colors $C_1,\ldots,C_m$ such that no two points of $S$ which are a (Euclidean) distance $d_{ij}$ apart are both colored $C_j$, for all $1\leq i \leq k$ and $1\leq j \leq m$. If no such integer exists then we say that ${\hatχ }^{(k)}(S)=\infty$. In this paper we show that ${\hatχ }^{(k)}(\mathbb{R})$ is finite for all $k$. |
| title | The $k^{\text th}$ Upper Chromatic Number of the Line |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2506.20772 |