Progressions in Euclidean Ramsey theory

TL;DR

This paper improves bounds on Euclidean Ramsey theory, demonstrating that certain colorings avoid specific collinear point configurations with large numbers of points, and extends results to various distances.

Contribution

It refines the maximum number of collinear points in colorings avoiding particular configurations, reducing the bound from 10^{50} to 1177, and explores similar results for different distances.

Findings

Red/blue colorings avoiding 3 red collinear points separated by unit distance exist with up to 1177 blue points.

The bounds on the number of blue collinear points are significantly reduced from previous results.

Results are extended to configurations with different distances between points.

Abstract

Conlon and Wu showed that there is a red/blue-coloring of $\mathbb{E}^n$ that does not contain $3$ red collinear points separated by unit distance and $m=10^{50}$ blue collinear points separated by unit distance. We prove that the statement holds with $m=1177$. We show similar results with different distances between the points.