Odd distances in colourings of the plane

TL;DR

This paper proves that in any finite coloring of the plane, there must exist two points of the same color separated by an odd distance, highlighting a fundamental property of plane colorings.

Contribution

The paper establishes a new result that guarantees the existence of monochromatic pairs at odd distances in any finite plane coloring, extending understanding of geometric coloring problems.

Findings

Monochromatic pairs at odd distances always exist in finite plane colorings

The result applies to all finite colorings of the plane

It advances the theory of geometric Ramsey problems

Abstract

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd distance from each other.