A Euclidean Ramsey result in the plane

TL;DR

This paper provides an elementary proof that in any red-blue coloring of the plane, there must exist either a red pair of points at unit distance or five blue points aligned with unit spacing.

Contribution

It offers a simple, elementary proof confirming a longstanding Euclidean Ramsey theory conjecture about colored points in the plane.

Findings

Confirmed the existence of a red pair at unit distance or five collinear blue points with unit spacing in any coloring.

Provided an elementary proof, simplifying previous complex approaches.

Strengthened understanding of Euclidean Ramsey properties in the plane.

Abstract

An old question in Euclidean Ramsey theory asks, if the points in the plane are red-blue coloured, does there always exist a red pair of points at unit distance or five blue points in line separated by unit distances? An elementary proof answers this question in affirmative.