Axiom of choice and chromatic number: Examples on the plane

In our previous paper (J. Combin. Theory Ser. A 103 (2) (2003) 387) we formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed an example of a distance graph on the real line R whose chromatic number depends upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct a distance graph G₂ on the plane R², thus coming much closer to the setting of the chromatic number of the plane problem. The chromatic number of G₂ is 4 in the Zermelo-Fraenkel-Choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. 92 Ser. 2 (1970) 1).