Total orders realizable as the distances between two sets of points
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918150980763648 |
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| author | Maldonado, Gerardo L. Pérez, Miguel Raggi Roldán-Pensado, Edgardo |
| author_facet | Maldonado, Gerardo L. Pérez, Miguel Raggi Roldán-Pensado, Edgardo |
| contents | In this note we give a negative answer to a question proposed by Almendra-Hernández and Martínez-Sandoval. Let $n\le m$ be positive integers and let $X$ and $Y$ be sets of sizes $n$ and $m$ in $\mathbb{R}^{n-1}$ such that every pair of points in $X\cup Y$ defines a unique distance. There is a natural order on $X\times Y$ induced by the distances between the corresponding points. The question is if all possible orders on $X\times Y$ can be obtained in this way. We show that the answer is negative when $n<m$. The case $n=m$ remains open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_05535 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Total orders realizable as the distances between two sets of points Maldonado, Gerardo L. Pérez, Miguel Raggi Roldán-Pensado, Edgardo Combinatorics Metric Geometry In this note we give a negative answer to a question proposed by Almendra-Hernández and Martínez-Sandoval. Let $n\le m$ be positive integers and let $X$ and $Y$ be sets of sizes $n$ and $m$ in $\mathbb{R}^{n-1}$ such that every pair of points in $X\cup Y$ defines a unique distance. There is a natural order on $X\times Y$ induced by the distances between the corresponding points. The question is if all possible orders on $X\times Y$ can be obtained in this way. We show that the answer is negative when $n<m$. The case $n=m$ remains open. |
| title | Total orders realizable as the distances between two sets of points |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2304.05535 |