Upper bound on the circular chromatic number of the plane (Q1753019)

Summary: 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 [0.9327,1.0673].