Canonical Ramsey: triangles, rectangles and beyond

In a seminal work, Cheng and Xu showed that if \(\mathcal{S}\) is a square or a triangle with a certain property, then for every positive integer \(r\) there exists \(n_0(\mathcal{S})\) independent of \(r\) such that every \(r\)-coloring of \(\mathbb{E}^n\) with \(n\ge n_0(\mathcal{S})\) contains a monochromatic or a rainbow congruent copy of \(\mathcal{S}\). Geh\'{e}r, Sagdeev, and T\'{o}th 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. \begin{enumerate}    \item  We resolve both questions. More precisely, for triangles we confirm the property in \(\mathbb{E}^4\) by developing a novel rotation-spherical 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.    \item Beyond this, we develop a concise perturbation framework based on an iterative embedding coupled with the Frankl–R\"{o}dl 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.\end{enumerate}