2025/08/04 by Gennian Ge, Shu Yang, Ge, Gennian +3
Computer Science · Mathematics · #05D10 #52C10 #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2508.02465
openalex publication_date 2025/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a seminal work, Cheng and Xu proved that for any positive integer \(r\), there exists an integer \(n0\), independent of \(r\), such that every \(r\)-coloring of the \(n\)-dimensional Euclidean space \(𝔼n\) with \(n ≥ n0\) 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 Geheér, Sagdeev, and Tóth, who extended the result to all hypercubes, and to rectangles whose side lengths \(a\), \(b\) satisfy \(((a)/(b))2\) 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 Geheér, Sagdeev, and Tóth. 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.