On four-colourings of the rational four-space (Q917570)

Let \(Q^ p\) denote the collection of all the rational points of the p- space \(E^ p\), and let \(G(Q^ p)\) denote the graph, obtained by taking \(Q^ p\) as the vertex set, and connecting 2 points iff they are at a distance 1 (the usual Euclidean distance). There is shown that in every 4-colouring of \(G(Q^ p)\) every colour is everywhere dense. The chromatic number of \(G(Q^ 6)\), \(G(Q^ 7)\) and \(G(Q^ 8)\) is at least 6, 8 and 9, respectively.