On colorings of a two-dimensional sphere partitioned into regions

In this paper we consider the problem of the chromatic number of the 2-dimensional sphere partitioned into regions, in the setting proposed by C. Thomassen. The focus is on «nice colorings», where regions are simply connected, have a diameter less than one, and any two regions of the same color are at a distance greater than one. An improved lower bound is established for the radius of the sphere beyond which a proper 7-coloring becomes impossible.