On the Nelson Unit Distance Coloring Problem

Abstract

In 1950 Nelson raised the problem of coloring the Euclidean plane in such a way that no two points of distance 1 receive the same color. How many colors are needed? This problem was often mentioned in Paul Erdos' famous lectures on unsolved combinatorial problems. The history of the problem is described in [2] and [3]. Clearly, three colors are needed. To see that four colors are needed, we consider seven points x1, X2,..., X7 in the Euclidean plane such that the following pairs are of distance 1: X1x2,x1x3,xXx4,x3x4,X2X5,X2X6,X5X6,X3X7,X4X7,X5X7, x6x7. It follows from the theorem of de Bruijn and Erdos [1] that the number of colors needed for the whole plane is the maximum number of colors needed for the finite subsets. The half-century old upper bound 7 is obtained by drawing an appropriate graph in the plane such that each face (region) is bounded by a cycle of (Euclidean) diameter less than one and then coloring each face and part of the boundary by the same color in such a way that only faces of distance > 1 receive the same color. We prove that colorings of this type always need at least 7 colors. More generally, 7 colors are needed for any surface and any metric of large diameter provided there are no short noncontractible curves and no short contractible curves whose interior have large area. The upper bound 7 is obtained from a hexagonal tiling of the plane such that the hexagons are regular and of diameter slightly less than 1. All points inside a hexagon are colored with the same color. Two hexagons are colored differently if the distance between them is less than one. A coloring of this type will be called nice. More generally, we consider any metric space S, d such that S is a surface, i.e., S is an arcwise connected Hausdorff space such that each element of S has a neighborhood homeomorphic to an open disc in the Euclidean plane. We let G be a connected graph on S, i.e., the vertices of G are elements of S, and the edges of G are simple arcs on S that are pairwise disjoint except at a common vertex. Moreover, we assume that each face (i.e., arcwise connected component of S\G) has diameter less than 1, is homeomorphic to a disc, and is bounded by a cycle in G. Now a nice coloring of S obtained from G is a coloring such that each color class is the union of faces (and part of their boundaries) such that the distance between any two of these faces is greater than 1. We define the area of subset A of S as the maximum number of pairwise disjoint open discs of radius 2 that are contained in A. (If this maximum does not exist we say that A has infinite area.) We say that a simple closed curve C is contractible if S\C has precisely two arcwise connected components such that one of them is homeomorphic to an open disc in the Euclidean plane. That component is called the interior of C and is denoted int(C). (If S is a sphere, then int(C) denotes any component of S\C of