# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1703.10723