Erdős, Paul; Guy, R. K. Distinct distances between lattice points. (English) Zbl 0222.10053 Elem. Math. 25, 121-123 (1970). Let \(k\) be the greatest number of points in real \(2\)-space with integer coordinates between \(1\) and \(n\) and for which all mutual distances are distinct. By a simple counting argument, \(k \leq n\). For \(2 \leq n \leq 7\), \(k=n\) is verified by a choice of points in the plane. From a result of E. Landau [Handbuch der Lehre von der Verteilung der Primzahlen (Leipzig, 1909), Bd. 2, p. 643] there is a positive constant \(c\) with \(k<cn(\log n)^{-1/4}\). A simple combinatorial proof is given that for \(\epsilon >0\), if \(n\) is sufficiently large, then \(k>n^{2/3-\epsilon}\). Results for dimensions \(1\) and \(3\) are mentioned. Two problems are suggested: 1. Find the minimum number of points, determining distinct distances, so that no point may be added without duplicating a distance. 2. Given any \(n\) points in the plane (or \(d\)-space) how many can one select so that the distances are all distinct? Reviewer: J.P.Tull Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 2 ReviewsCited in 4 Documents MSC: 11N56 Rate of growth of arithmetic functions PDFBibTeX XMLCite \textit{P. Erdős} and \textit{R. K. Guy}, Elem. Math. 25, 121--123 (1970; Zbl 0222.10053) Full Text: EuDML Online Encyclopedia of Integer Sequences: Numerators of the squared radii of the smallest enclosing circles of n points with integer coordinates and distinct mutual distances, arranged such that the radius of their enclosing circle is minimized. Denominators are given in A193556. Size k of smallest square of k X k lattice points from which n points with distinct mutual distances can be chosen. Size of maximal subset of points of n X n grid such that no two points are at the same distance. Number of largest subsets of the set of points in an n X n square grid, such that no two points are at the same distance.