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arXiv · 2209.13247

Euclidean Gallai-Ramsey Theory

Yaping Mao·Kenta Oeki·Zhao Wang
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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 $K'$, does there exist an integer $n_0$ such that for any $r$-coloring of the points of $n$-dimensional Euclidean space with $n \geq n_0$, there is a monochromatic configuration congruent to $K$ or a rainbow configuration congruent to $K'$? In particular, we give a bound on $n_0$ for some configurations $K$ and $K'$, such as triangles and rectangles. Those are extensions of ordinary Euclidean Ramsey theory where the purpose is to find a monochromatic configuration.

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Yaping Mao, Kenta Oeki, Zhao Wang. 2022-09-27. Euclidean Gallai-Ramsey Theory. https://arxiv.org/abs/2209.13247

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