Avoiding short progressions in Euclidean Ramsey theory
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914101311045632 |
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| author | Currier, Gabriel Moore, Kenneth Yip, Chi Hoi |
| author_facet | Currier, Gabriel Moore, Kenneth Yip, Chi Hoi |
| contents | We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if $\ell_m$ denotes $m$ collinear points with consecutive points of distance one apart, we say that $\mathbb{E}^n \not \to (\ell_r,\ell_s)$ if there is a red/blue coloring of $n$-dimensional Euclidean space that avoids red congruent copies of $\ell_r$ and blue congruent copies of $\ell_s$. We show that $\mathbb{E}^n \not \to (\ell_3, \ell_{20})$, improving the best-known result $\mathbb{E}^n \not \to (\ell_3, \ell_{1177})$ by Führer and Tóth, and also establish $\mathbb{E}^n \not \to (\ell_4, \ell_{14})$ and $\mathbb{E}^n \not \to (\ell_5, \ell_{8})$ in the spirit of the classical result $\mathbb{E}^n \not \to (\ell_6, \ell_{6})$ due to Erdős et. al. We also show a number of similar $3$-coloring results, as well as $\mathbb{E}^n \not \to (\ell_3, α\ell_{6889})$, where $α$ is an arbitrary positive real number. This final result answers a question of Führer and Tóth in the positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Avoiding short progressions in Euclidean Ramsey theory Currier, Gabriel Moore, Kenneth Yip, Chi Hoi Combinatorics Number Theory 05D10, 52C10, 11B25 We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if $\ell_m$ denotes $m$ collinear points with consecutive points of distance one apart, we say that $\mathbb{E}^n \not \to (\ell_r,\ell_s)$ if there is a red/blue coloring of $n$-dimensional Euclidean space that avoids red congruent copies of $\ell_r$ and blue congruent copies of $\ell_s$. We show that $\mathbb{E}^n \not \to (\ell_3, \ell_{20})$, improving the best-known result $\mathbb{E}^n \not \to (\ell_3, \ell_{1177})$ by Führer and Tóth, and also establish $\mathbb{E}^n \not \to (\ell_4, \ell_{14})$ and $\mathbb{E}^n \not \to (\ell_5, \ell_{8})$ in the spirit of the classical result $\mathbb{E}^n \not \to (\ell_6, \ell_{6})$ due to Erdős et. al. We also show a number of similar $3$-coloring results, as well as $\mathbb{E}^n \not \to (\ell_3, α\ell_{6889})$, where $α$ is an arbitrary positive real number. This final result answers a question of Führer and Tóth in the positive. |
| title | Avoiding short progressions in Euclidean Ramsey theory |
| topic | Combinatorics Number Theory 05D10, 52C10, 11B25 |
| url | https://arxiv.org/abs/2404.19233 |