First non-trivial upper bound on the circular chromatic number of the plane

TL;DR

This paper establishes the first non-trivial upper bound of approximately 6.30 for the circular chromatic number of the plane, improving upon the previous bound of 7, in the context of a circular coloring variant of the Nelson-Hadwiger problem.

Contribution

It provides the first known circular coloring of the plane with a perimeter less than 7, specifically around 6.30, for points at distance one, advancing the understanding of the plane's chromatic properties.

Findings

Existence of an $r$-circular coloring with $r \, \approx \, 6.30$

First such coloring with $r$ less than 7

Coloring for specific distance intervals within the plane

Abstract

We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In $r$-circular coloring we assign arcs of length one of a circle with a perimeter $r$ in such a way that points at distance one get disjoint arcs. In this paper we show the existence of $r$-circular coloring for $r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circular coloring of the plane with $r$ smaller than 7. We also show $r$-circular coloring of the plane with $r<7$ in the case when we require disjoint arcs for points at distance belonging to the internal $[\frac{10}{11}, \frac{12}{11}]$.