Unit distance graphs with ambiguous chromatic number

TL;DR

This paper constructs a new class of infinite unit distance graphs with ambiguous chromatic number, providing evidence that the chromatic number of the plane may depend on set-theoretic axioms.

Contribution

It introduces a novel class of unit distance graphs with ambiguous chromatic number, expanding understanding of set-theoretic influences on graph coloring.

Findings

Existence of unit distance graphs with ambiguous chromatic number

Potential implications for the chromatic number of the plane

Supports set-theoretic dependence of graph coloring properties

Abstract

First Laszlo Szekely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with vertex set R^n, and hence may be seen as further evidence that the chromatic number of the plane might depend on set theory.