All rectangles exhibit canonical Ramsey property

TL;DR

This paper proves that every rectangle in Euclidean space has the canonical Ramsey property, extending the phenomenon of dimension-independence to all rectangles regardless of aspect ratio.

Contribution

It resolves the open problem by showing all rectangles exhibit the canonical Ramsey property, using a new structural reduction and advanced Ramsey theorems.

Findings

All rectangles exhibit the canonical Ramsey property.

Introduces a structural reduction linking product configurations to Ramsey theorems.

Enables control of arbitrary aspect ratios in Euclidean space.

Abstract

In a seminal work, Cheng and Xu proved that for any positive integer \(r\), there exists an integer \(n_0\), independent of \(r\), such that every \(r\)-coloring of the \(n\)-dimensional Euclidean space \(\mathbb{E}^n\) with \(n \ge n_0\) contains either a monochromatic or a rainbow congruent copy of a square. This phenomenon of dimension-independence was later formalized as the canonical Ramsey property by Gehe\'{e}r, Sagdeev, and T\'{o}th, who extended the result to all hypercubes, and to rectangles whose side lengths \(a\), \(b\) satisfy \((\frac{a}{b})^2\) is rational. They further posed the natural problem of whether every rectangle admits the canonical Ramsey property, regardless of the aspect ratio. In this paper, we show that all rectangles exhibit the canonical Ramsey property, thereby completely resolving this open problem of Gehe\'{e}r, Sagdeev, and T\'{o}th. Our proof introduces a new structural reduction that identifies product configurations with bounded color complexity, enabling the application of simplex Ramsey theorems and product Ramsey amplification to control arbitrary aspect ratios.