On existence of integral point sets and their diameter bounds
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arXiv
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866908682273423360 |
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| author | Avdeev, Nikolai |
| author_facet | Avdeev, Nikolai |
| contents | A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower bound for diameter of planar integral point sets. This improvement takes into account some results related to the Point Packing in a Square problem. Then for arbitrary integers $m \geq 2$, $n \geq m+1$, $d \geq 1$ we give a construction of an integral point set $M$ of $n$ points in $m$-dimensional Euclidean space, where $M$ contains points $M_1$ and $M_2$ such that distance between $M_1$ and $M_2$ is exactly $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_11926 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On existence of integral point sets and their diameter bounds Avdeev, Nikolai Combinatorics 52C10 A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower bound for diameter of planar integral point sets. This improvement takes into account some results related to the Point Packing in a Square problem. Then for arbitrary integers $m \geq 2$, $n \geq m+1$, $d \geq 1$ we give a construction of an integral point set $M$ of $n$ points in $m$-dimensional Euclidean space, where $M$ contains points $M_1$ and $M_2$ such that distance between $M_1$ and $M_2$ is exactly $d$. |
| title | On existence of integral point sets and their diameter bounds |
| topic | Combinatorics 52C10 |
| url | https://arxiv.org/abs/1906.11926 |