Uniform distances in rational unit-distance graphs

Abstract Let G be the graph obtained from all the rational points in the d -space E d by connecting every pair at Euclidean distance one. It is known that G is a connected graph, provided d ⩾ 5. We establish an inequality of the form dist G ( x, y ) ⩽ ⌈| x − y |⌉ + 1, for all d ⩾ 8, between the Euclidean distance | x − y | of any two rational points x and y and their corresponding distance dist G ( x, y ) in the graph G . A slightly weaker relation is shown to hold in dimensions 5, 6 and 7.