Monochromatic unit equilateral triangle on low-dimensional spheres
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918506509893632 |
|---|---|
| author | Zhao, Xiaochen Ge, Gennian |
| author_facet | Zhao, Xiaochen Ge, Gennian |
| contents | A result of Matoušek and Rödl in 1995 states that for every $\varepsilon>0$ and every triangle $T$ with circumradius $ρ(T)$, there exists a dimension $n=n(\varepsilon,T)$ such that every $2$-coloring of the $n$-dimensional sphere of radius $ρ(T)+\varepsilon$, namely $\mathbb{S}^{n}(ρ(T)+\varepsilon)$, contains a monochromatic congruent copy of $T$. In this paper, we determine the exact threshold dimension for the unit equilateral triangle on the sphere $\mathbb{S}^{n}(1/\sqrt{2})$: there exists a $2$-coloring of $\mathbb{S}^{2}(1/\sqrt{2})$ with no monochromatic unit equilateral triangle, whereas every $2$-coloring of $\mathbb{S}^{3}(1/\sqrt{2})$ contains one. Along the way, we also establish several further Euclidean Ramsey-type results on low-dimensional spheres, including asymmetric and isosceles variants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16958 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monochromatic unit equilateral triangle on low-dimensional spheres Zhao, Xiaochen Ge, Gennian Combinatorics Metric Geometry 05D10(Primary) 52C10, 51K99 (Secondary) A result of Matoušek and Rödl in 1995 states that for every $\varepsilon>0$ and every triangle $T$ with circumradius $ρ(T)$, there exists a dimension $n=n(\varepsilon,T)$ such that every $2$-coloring of the $n$-dimensional sphere of radius $ρ(T)+\varepsilon$, namely $\mathbb{S}^{n}(ρ(T)+\varepsilon)$, contains a monochromatic congruent copy of $T$. In this paper, we determine the exact threshold dimension for the unit equilateral triangle on the sphere $\mathbb{S}^{n}(1/\sqrt{2})$: there exists a $2$-coloring of $\mathbb{S}^{2}(1/\sqrt{2})$ with no monochromatic unit equilateral triangle, whereas every $2$-coloring of $\mathbb{S}^{3}(1/\sqrt{2})$ contains one. Along the way, we also establish several further Euclidean Ramsey-type results on low-dimensional spheres, including asymmetric and isosceles variants. |
| title | Monochromatic unit equilateral triangle on low-dimensional spheres |
| topic | Combinatorics Metric Geometry 05D10(Primary) 52C10, 51K99 (Secondary) |
| url | https://arxiv.org/abs/2605.16958 |