Euclidean Gallai-Ramsey for various configurations

TL;DR

This paper investigates the Euclidean Gallai-Ramsey problem in multi-dimensional spaces, focusing on various geometric configurations and revealing that space dimensions often do not depend on the number of colors, with proofs mainly geometric.

Contribution

It extends Gallai-Ramsey results to new configurations in Euclidean spaces and shows dimension independence from the number of colors, refining previous findings.

Findings

Dimensions often independent of number of colors

Refined results for triangles, squares, lines, and special configurations

Geometric proofs underpin the results

Abstract

The Euclidean Gallai-Ramsey problem, which investigates the existence of monochromatic or rainbow configurations in a colored $n$-dimensional Euclidean space $\mathbb{E}^{n}$, was introduced and studied recently. We further explore this problem for various configurations including triangles, squares, lines, and the structures with specific properties, such as rectangular and spherical configurations. Several of our new results provide refinements to the results presented in a recent work by Mao, Ozeki and Wang. One intriguing phenomenon evident on the Gallai-Ramsey results proven in this paper is that the dimensions of spaces are often independent of the number of colors. Our proofs primarily adopt a geometric perspective.