Canonical Ramsey: triangles, rectangles and beyond

TL;DR

The paper resolves two central open questions in Euclidean Canonical Ramsey theory: whether all triangles and all rectangles possess the canonical Ramsey property. It proves that in dimension four, every $r$-coloring of $\mathbb{E}^4$ yields either a monochromatic or rainbow congruent copy of any fixed triangle, and it shows that for rectangles, a high-dimensional canonical Ramsey property holds via a structural reduction to bounded color complexity and a fusion of the Frankl–Rödl simplex Ramsey theorem with product Ramsey theory. Beyond these results, the authors develop a concise perturbation framework based on the simplex super-Ramsey theorem, enabling canonical Ramsey results for a natural class of $3$-dimensional simplices and providing an alternative proof pathway for triangles. The methods combine rotation-spherical chaining, bounded-color-structure reductions, and high-dimensional perturbations, offering tools likely to influence further directions in canonical and Euclidean Ramsey theory and its connections with simplex Ramsey phenomena.

Abstract

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.

Figures (7)

• Figure 2.1: The fifth point $N$
• Figure 2.2: Case 1
• Figure 2.3: Case 2
• Figure 2.5: Notice that $\chi(C)=\chi(D)=$red since $\mathcal{S}$ is red, then $\chi(Z)\neq$ red for $\triangle ZCD \cong \mathcal{T}$, which implies that all points in the sphere $\mathcal{W}$ are not colored red.
• Figure 2.6: The left figure illustrates the monochromatic sphere determined by $P$ and $Q$ intersects both the red and non-red sphere. The right figure shows their projection on $\mathbb{Y}$, which is the perpendicular bisector of segment $PQ$.

Theorems & Definitions (55)

• Theorem 1.2
• Theorem 1.3
• Definition 1.4: $\varepsilon$-sphere
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1
• proof : Proof of \ref{['prop:KeyInequality']}
• Proposition 2.2: 2025DCGChengXu
• Lemma 2.3
• proof : Proof of \ref{['lem:5-point structure']}