Another six-coloring of the plane

A six-coloring of the Euclidean plane is constructed such that the distance 1 is not realized by any color except one, which does not realize the distance x/2 1. A 44-year-old problem due to Edward Nelson asks to find the chromatic number x(E 2) of the plane, i.e. the minimal number of colors that are required for coloring the plane so that no two points, a unit distance apart, lie on the same color. (For a history of the problem we refer the reader to [3].) It has been known, since 1950, that 4 ~< z(E 2) ~< 7. Related questions have been considered (see [1] for a number of them). We would like to pose a new problem. Almost chromatic number za(E 2) of the plane is the minimal number of colors that are required for coloring the plane so that almost all (i.e. all but one) colors forbid the unit distance, and the remaining color forbids a distance. We have the following inequalities for Xa(E2): 4 <~ za(E 2) <~ 6. The lower bound follows from Dmitry Raiskii's [2]. The upper bound was proven by the second author in [4]. The problem now is: find z~(E 2). * Corresponding author. 0012-365X/96/$15.00 © 1996--Elsevier Science B.V. All rights reserved SSDI 0012-365X(95)00210-3 428 1. Hoffman, A. Soifer/ Discrete Mathematics 150 (1996) 427-429