Point sets with distinct distances

Abstract

For positive integersd andn letf d (n) denote the maximum cardinality of a subset of then d-gird {1,2,...,n}d with distinct mutual euclidean distances. Improving earlier results of Erdős and Guy, it will be shown thatf 2 (n)≥c·n 2/3 and, ford≥3, thatf d (n)≥c d ·n 2/3 ·(lnn)1/3, wherec, c d >0 are constants. Also improvements of lower bounds of Erdős and Alon on the size of Sidon-sets in {12,222,...,n 2} are given.Furthermore, it will be proven that any set ofn points in the plane contains a subset with distinct mutual distances of sizec 1·n 1/4, and for point sets in genral position, i.e. no three points on a line, of sizec 2·n 1/3 with constantsc 1,c 2>0. To do so, it will be shown that forn points in ℝ2 with distinct distancesd 1,d 2,...,d t , whered i has multiplicitym i , one has ∑ t i=1 m 2 i ≤c·n 3.25 for a positive constantc. If then points are in general position, then we prove ∑ t i=1 m 2 i ≤c·n 3 for a positive constantc 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.