The Ramsey-to-canonical Ramsey conjecture for configurations

Let SEn\mathcal{S}\subseteq\mathbb{E}^n be a configuration. It is Ramsey if for every rNr\in\mathbb{N} there exists a dimension n=n(S,r)n=n(\mathcal{S},r) such that EnrS\mathbb{E}^n\overset{r}{\rightarrow}\mathcal{S}. A configuration exhibits the canonical Ramsey property if, for every rNr\in\mathbb{N} and all sufficiently large dimensions, every rr-coloring of the ambient Euclidean space contains a monochromatic or rainbow congruent copy of S\mathcal{S}. Ramsey-to-canonical Ramsey conjecture. If S\mathcal{S} is Ramsey, then S\mathcal{S} exhibits the canonical Ramsey property. This would unify the known canonical Ramsey results for rectangles, triangles, and certain simplices; determining whether all simplices have the canonical Ramsey property remains open.

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Primary source

Yijia Fang, Gennian Ge, Yang Shu, Qian Xu, Zixiang Xu and Dilong Yang, “Canonical Ramsey: triangles, rectangles and beyond”, arXiv:2510.11638 (2025).

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