Axiom of choice and chromatic number of Rn

In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I 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 examples of a distance graph on the real line R and difference graphs on the real plane R2 whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space Rn, whose chromatic number is a positive integer 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. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory.