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 Geheér, Sagdeev, and Tóth, 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 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.
Canonical Ramsey: triangles, rectangles and beyond
National University of Singapore（新加坡国立大学）；Capital Normal University（首都师范大学）；University of Science and Technology of China（中国科学技术大学）；Peking University（北京大学）；Institute for Basic Science (IBS)（韩国基础科学研究院）
Ramakrishna Mission Vivekananda Educational and Research Institute（罗摩克里希纳使命维韦卡南达教育与研究机构）；Centre de Recerca Matemàtica（数学研究中心）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
Ramsey expansions of metrically homogeneous graphs
Charles University（查理大学）；Czech Academy of Sciences（捷克科学院）；National and Kapodistrian University of Athens（雅典国立卡波季斯特里安大学）；TU Dresden（德累斯顿工业大学）；University of Toronto Mississauga（多伦多大学密西沙加分校）
