On the space chromatic number
A nice open problem is to find the minimum number of colors necessary to color the points of the Euclidean plane so that any two points of unit distance receive distinct colors. This number is known to be at least 4 (by finding a particular set of 6 points that require 4 colors) and 7 (by demonstrating a particular coloring of all the points). It is related to finding the maximum chromatic number of unit distance graphs: those graphs that immerse in the plane so that all edges are line segments of unit length. The problem generalizes nicely to all metric spaces, in particular, to \(d\)-dimensional Euclidean space. In this paper the author examines the chromatic number of 3-dimensional Euclidean space. He gives an example that shows the chromatic number is at least 6, improving the previously best known lower bound of 5. The best known upper bound is 21, an impressively large gap. He also gives a Ramsey-type result for 2-coloring 3-space.
scientific article; zbMATH DE number 1838948
scientific article