Non-spherical sets versus lines in Euclidean Ramsey theory
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908525948567552 |
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| author | Conlon, David Führer, Jakob |
| author_facet | Conlon, David Führer, Jakob |
| contents | We show that for every non-spherical set $X$ in $\mathbb{E}^d$, there exists a natural number $m$ and a red/blue-colouring of $\mathbb{E}^n$ for every $n$ such that there is no red copy of X and no blue progression of length $m$ with each consecutive point at distance $1$. This verifies a conjecture of Wu and the first author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07718 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-spherical sets versus lines in Euclidean Ramsey theory Conlon, David Führer, Jakob Combinatorics 05D10, 52C10 We show that for every non-spherical set $X$ in $\mathbb{E}^d$, there exists a natural number $m$ and a red/blue-colouring of $\mathbb{E}^n$ for every $n$ such that there is no red copy of X and no blue progression of length $m$ with each consecutive point at distance $1$. This verifies a conjecture of Wu and the first author. |
| title | Non-spherical sets versus lines in Euclidean Ramsey theory |
| topic | Combinatorics 05D10, 52C10 |
| url | https://arxiv.org/abs/2406.07718 |