Equally spaced collinear points in Euclidean Ramsey theory

TL;DR

This paper proves a new Euclidean Ramsey theory result showing that in any red-blue colouring of k-dimensional space, either two red points are unit distance apart or there are k+3 blue collinear points with consecutive points at unit distance, for 4≤k≤10.

Contribution

It establishes a novel Euclidean Ramsey theorem for dimensions 4 to 10, identifying specific monochromatic configurations guaranteed by any two-coloring.

Findings

Either two red points are distance one apart or a blue collinear chain of length k+3 exists.

The result is new for dimensions 4 through 10.

Provides bounds for monochromatic configurations in Euclidean space.

Abstract

It is proved that for $k\geq 4$, if the points of $k$-dimensional Euclidean space are coloured in red and blue, then there are either two red points distance one apart or $k+3$ blue collinear points with distance one between any two consecutive points. This result is new for $4\leq k\leq 10$.