Nearly all known Euclidean Ramsey sets are subsoluble
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908692821049344 |
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| author | Behague, Natalie |
| author_facet | Behague, Natalie |
| contents | A finite set $X$ in a Euclidean space $\mathbb{R}^d$ is called Ramsey if for every $k$ there exists an integer $n$ such that whenever $\mathbb{R}^n$ is coloured with $k$ colours, there is a monochromatic copy of $X$. Graham conjectured that all spherical sets are Ramsey, but progress on this conjecture has been slow. A key result of Kříž is that all sets that embed in sets that are acted on transitively by a soluble group are Ramsey. We show that for nearly all known examples of Ramsey sets the converse is true, with only two possible exceptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15677 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nearly all known Euclidean Ramsey sets are subsoluble Behague, Natalie Combinatorics A finite set $X$ in a Euclidean space $\mathbb{R}^d$ is called Ramsey if for every $k$ there exists an integer $n$ such that whenever $\mathbb{R}^n$ is coloured with $k$ colours, there is a monochromatic copy of $X$. Graham conjectured that all spherical sets are Ramsey, but progress on this conjecture has been slow. A key result of Kříž is that all sets that embed in sets that are acted on transitively by a soluble group are Ramsey. We show that for nearly all known examples of Ramsey sets the converse is true, with only two possible exceptions. |
| title | Nearly all known Euclidean Ramsey sets are subsoluble |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.15677 |