Point sets with distinct distances

For positive integersd andn letfd(n) denote the maximum cardinality of a subset of thend-gird {1,2,...,n}d with distinct mutual euclidean distances. Improving earlier results of Erdős and Guy, it will be shown thatf2(n)≥c·n2/3 and, ford≥3, thatfd(n)≥cd·n2/3 ·(lnn)1/3, wherec, cd>0 are constants. Also improvements of lower bounds of Erdős and Alon on the size of Sidon-sets in {12,222,...,n2} are given.Furthermore, it will be proven that any set ofn points in the plane contains a subset with distinct mutual distances of sizec1·n1/4, and for point sets in genral position, i.e. no three points on a line, of sizec2·n1/3 with constantsc1,c2>0. To do so, it will be shown that forn points in ℝ2 with distinct distancesd1,d2,...,dt, wheredi has multiplicitymi, one has ∑i=1tmi2≤c·n3.25 for a positive constantc. If then points are in general position, then we prove ∑i=1tmi2≤c·n3 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.