Axiom of choice and chromatic number of the plane

Abstract

Define a graph U on the set of all points of the plane R as its vertex set, with two points adjacent iff they are distance 1 apart. The graph U ought to be called unit distance plane, and its chromatic number w is called chromatic number of the plane. Finite subgraphs of U are called finite unit distance plane graphs. In 1950 the 18-year old Edward Nelson posed the problem of finding w (see the problem’s history in [Soi1]). A number of relevant results were obtained under additional restrictions on monochromatic sets (see surveys in [CFG,KW,Soi2,Soi3]). Falconer, for example, showed [F] that w is at least 5 if monochromatic sets are Lebesgue measurable. Amazingly though, the problem has withstood all assaults in the general case, leaving us with an embarrassingly wide range for w being 4, 5, 6 or 7. In their fundamental 1951 paper [EB], Erdos and de Bruijn have shown that the chromatic number of the plane is attained on some finite subgraph. This result has naturally channeled much of research in the direction of finite unit distance graphs. One limitation of the Erdos–de-Bruijn result, however, has remained a low key: they ARTICLE IN PRESS