Canonical Ramsey: triangles, rectangles and beyond
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911208337047552 |
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| author | Fang, Yijia Ge, Gennian Shu, Yang Xu, Qian Xu, Zixiang Yang, Dilong |
| author_facet | Fang, Yijia Ge, Gennian Shu, Yang Xu, Qian Xu, Zixiang Yang, Dilong |
| contents | In a seminal work, Cheng and Xu showed that if $S$ is a square or a triangle with a certain property, then for every positive integer $r$ there exists $n_0(S)$ independent of $r$ such that every $r$-coloring of $\mathbb{E}^n$ with $n\ge n_0(S)$ contains a monochromatic or a rainbow congruent copy of $S$. Gehér, Sagdeev, and Tóth formalized this dimension independence as the canonical Ramsey property and proved it for all hypercubes, thereby covering rectangles whose squared aspect ratio $(a/b)^2$ is rational. They asked whether this property holds for all triangles and for all rectangles.
(1) We resolve both questions. More precisely, for triangles we confirm the property in $\mathbb{E}^4$ by developing a novel rotation-sphereical chaining argument. For rectangles, we introduce a structural reduction to product configurations of bounded color complexity, enabling the use of the simplex Ramsey theorem together with product Ramsey theorem.
(2) Beyond this, we develop a concise perturbation framework based on an iterative embedding coupled with the Frankl-Rödl simplex super-Ramsey theorem, which yields the canonical Ramsey property for a natural class of 3-dimensional simplices and also furnishes an alternative proof for triangles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canonical Ramsey: triangles, rectangles and beyond Fang, Yijia Ge, Gennian Shu, Yang Xu, Qian Xu, Zixiang Yang, Dilong Combinatorics Metric Geometry 52C10, 05D10 In a seminal work, Cheng and Xu showed that if $S$ is a square or a triangle with a certain property, then for every positive integer $r$ there exists $n_0(S)$ independent of $r$ such that every $r$-coloring of $\mathbb{E}^n$ with $n\ge n_0(S)$ contains a monochromatic or a rainbow congruent copy of $S$. Gehér, Sagdeev, and Tóth formalized this dimension independence as the canonical Ramsey property and proved it for all hypercubes, thereby covering rectangles whose squared aspect ratio $(a/b)^2$ is rational. They asked whether this property holds for all triangles and for all rectangles. (1) We resolve both questions. More precisely, for triangles we confirm the property in $\mathbb{E}^4$ by developing a novel rotation-sphereical chaining argument. For rectangles, we introduce a structural reduction to product configurations of bounded color complexity, enabling the use of the simplex Ramsey theorem together with product Ramsey theorem. (2) Beyond this, we develop a concise perturbation framework based on an iterative embedding coupled with the Frankl-Rödl simplex super-Ramsey theorem, which yields the canonical Ramsey property for a natural class of 3-dimensional simplices and also furnishes an alternative proof for triangles. |
| title | Canonical Ramsey: triangles, rectangles and beyond |
| topic | Combinatorics Metric Geometry 52C10, 05D10 |
| url | https://arxiv.org/abs/2510.11638 |