# Lines in Euclidean Ramsey theory

## TL;DR

This paper proves the existence of colorings in Euclidean spaces that avoid certain line configurations, establishing bounds related to Erdős et al.'s 1973 question about monochromatic line segments.

## Contribution

It demonstrates the existence of colorings in high-dimensional Euclidean spaces avoiding specific line patterns, answering a longstanding open problem.

## Key findings

- Existence of colorings avoiding red $ ext{ell}_2$ and blue $ ext{ell}_m$ for large m
- Bounds are tight up to a constant in the exponent
- Addresses a question posed by Erdős et al. in 1973

## Abstract

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $\ell_m$ for any $m \geq 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number $n$, there is a set $K \subset \mathbb{E}^1$ and a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $K$.