A result in asymmetric Euclidean Ramsey theory

TL;DR

This paper proves a new result in Euclidean Ramsey theory, showing that in any red-blue colouring of three-dimensional space, certain monochromatic configurations must exist, specifically either a red pair at unit distance or a blue collinear chain of six points with unit spacing.

Contribution

It introduces a novel Euclidean Ramsey theorem for three-dimensional space, establishing the existence of specific monochromatic configurations under any two-colouring.

Findings

Existence of a red pair at unit distance in any 2-colouring of 3D space.

Existence of six collinear blue points with consecutive points at unit distance.

The result extends Euclidean Ramsey theory to new geometric configurations.

Abstract

It is proved that if the points of the three-dimensional Euclidean space are coloured in red and blue, then there exist either two red points unit distance apart, or six collinear blue points with distance one between any two consecutive points.