On \(\surd\mathbf Q\)-distances (Q1911846)

A subset of a Euclidean space is said to be a \(\sqrt Q\)-set if, for every two points of the subset, the square of the distance between them is rational. Let \(s(n, N)\) denote the minimum number of pairs of points that need to be checked, to confirm that \(N\) points in general position in \(\mathbb{R}^n\) do form a \(\sqrt Q\)-set. The author finds \(s(n, N)\) exactly, for \(N\leq n+ 4\), and gives upper and lower bounds in general. The concept of \(n\)-valid graph is used to obtain these results.