On the chromatic number of the plane for map-type colorings
Georgy Sokolov, Vsevolod Voronov
TL;DR
This work studies a map-type variant of the Hadwiger–Nelson plane coloring problem, aiming to determine the chromatic number of the plane under map-like partitioning constraints. It develops a local-geometry toolkit—covering unit circles, three-color stripes around triple-boundary points, circle pseudocolorings, and complementary curves with an index—to derive obstructions to 6-color configurations. The main results establish a lower bound of $\\chi_{map}(\\mathbb{R}^2) \\ge 7$ under natural assumptions, and show how these bounds extend to polygonal maps and related variants via a forbidden-distance framework. Collectively, the paper advances the program toward a definitive value for the map-chromatic number of the plane, highlighting structural constraints that rule out 6-color map colorings in the considered regimes.
Abstract
We consider the Hadwiger-Nelson problem on the chromatic number of the plane under conditions of coloring a map containing a finite number of vertices in any bounded region. Woodall (1973) and Townsend (1981) showed that at least 6 colors would be required. In the present paper, it is shown that at least 7 colors are required to color a map in which the boundaries are not arcs of a unit circle and three boundaries connect at each vertex. As a corollary, we obtain that at least 7 colors are required for a proper coloring in which the regions are arbitrary polygons. The proof relies on techniques developed for a similar result concerning the chromatic number of the plane with a forbidden interval of distances.