On the existence of 4-chromatic subgraphs of G(ℚ3, d)

We address the chromatic number of a class of Euclidean distance graphs having vertex set Q, the 3-dimensional rational space. It is shown that if s is any positive integer with no prime factors congruent to 2 (mod 3), and G is the graph with vertex set Q where any two vertices are adjacent if and only if they are distance √ 2s apart, then G has chromatic number 4. Along the way, we obtain a few results on the chromatic numbers of certain Euclidean distance graphs having vertex set Z, the 3-dimensional integer space. We conclude by constructing an example (possibly the first) of a triangle-free, 4-chromatic distance graph in Q.