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We consider circular version of the famous Nelson-Hadwiger problem. It isknow that 4 colors are necessary and 7 colors suffice to color the euclideanplane 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 paperwe show the existence of $r$-circular coloring for$r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circularcoloring of the plane with $r$ smaller than 7. We also show $r$-circularcoloring of the plane with $r<7$ in the case when we require disjoint arcs forpoints at distance belonging to the internal $[\frac{10}{11}, \frac{12}{11}]$.

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