Patch Colorings and Rigid Colorings of the Rational n-Space

We investigate k-colorings of the rational n-space, Qn, such that any two points at distance one get distinct colors. Two types of colorings are considered: patch colorings where the colors occupy open sets with parts of their boundary, and rigid colorings which uniquely extend from any open subset of Qn. We prove that the existence of a patch k-coloring of Qn implies the existence of a k-coloring of Rn. We show that every 2-coloring of Q2 or Q3, and every 4-coloring of Q4 is rigid. We also construct a rigid 3-coloring of Q2.