POINT SETS WITH DISTINCT DISTANCES

For positive integers d and n let f(d)(n) denote the maximum cardinality of a subset of the n(d)-grid {1, 2, ..., n}(d) with distinct mutual euclidean distances. Improving earlier results of Erdos and Guy, it will be shown that f(2)(n) greater than or equal to c . n(2/3) and, for d greater than or equal to 3, that f(d)(n) greater than or equal to c(d) . n(2/3) . (ln n (1/3), where c, c(d) > 0 are constants. Also improvements of lower bounds of Erdos and Alon on the size of Sidon-sets in (1(2), 2(2), ..., n(2)) are given. Furthermore, it will be proven that any set of n points in the plane contains a subset with distinct mutual distances of size c(1) . n(1/4), and for point sets in general position, i.e. no three points on a line, of size c(2) . n(1/3) with constants c(1), c(2) > 0. To do so, it will be shown that for n points in R(2) with distinct distances d(1), d(2), ..., d(t), where d(i) has multiplicity m(i), one has Sigma(i=1)(t) m(i)(2) less than or equal to c . n(3.25) for a positive constant c. If the n points are in general position, then we prove Sigma(i=1)(t) m(i)(2) less than or equal to c . n(3) for a positive constant c and this bound is tight. Moreover, we give an efficient sequential algorithm for finding a subset of a given set with the desired properties, for example with distinct distances, of size as guaranteed by the probabilistic method under a more general setting.