DISTANCES FORBIDDEN BY 2-COLORINGS OF Q3 AND AN

For X = Q3 or A(n) (where A(n) is the set of points in Q(n) whose coordinates have odd denominators), we characterize all sets of distances D subset-of R+ with the following property: there exists some two-coloring of X such that, for all d is-an-element-of D, no two points in X that are a distance d apart are the same color. We also find all numbers d0 is-an-element-of R+ such that all sets of distances D subset-of R+ with this property retain the property under multiplication or division by d0.