More on lines in Euclidean Ramsey theory

TL;DR

This paper investigates Euclidean Ramsey theory, demonstrating that for certain sequences of points on a line, there exist colorings of Euclidean space avoiding specific monochromatic configurations.

Contribution

It proves the existence of colorings in Euclidean space that avoid monochromatic copies of certain point sequences, answering open questions in the field.

Findings

Existence of colorings avoiding red $ ext{l}_3$ and blue $ ext{l}_m$

For every dimension, such colorings can be constructed

Answers open questions in Euclidean Ramsey theory

Abstract

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points at distance one. Answering a question raised by Fox and the first author and independently by Arman and Tsaturian, we show that there is a natural number $m$ and a red/blue-colouring of $\mathbb{E}^n$ for every $n$ that contains no red copy of $\ell_3$ and no blue copy of $\ell_m$.