Euclidean Gallai-Ramsey Theory

Abstract

In this paper, we introduce Euclidean Gallai-Ramsey theory, by combining Euclidean Ramsey theory and Gallai-Ramsey theory on graphs. More precisely, we consider the following problem: For an integer $r$ and configurations $K$ and $K^{\prime }$, does there exist an integer $n_0$ such that for any $r$-coloring of the points of $n$-dimensional Euclidean space with $n\ge n_0$, there is a monochromatic configuration congruent to $K$ or a rainbow configuration congruent to $K^{\prime }$? In particular, we give a bound on $n_0$ for some configurations $K$ and $K^{\prime }$, such as triangles and rectangles. Those are extensions of ordinary Euclidean Ramsey theory where the purpose is to find a monochromatic configuration.