A set of points in the plane is said to be in general position if no three of them are collinear and no four of them are cocircular. If a point set determines only distinct vectors, it
is called parallelogram free. We show that there exist n-element point sets in the plane in general position, and parallelogram free, that determine only O(n2/√log n) distinct distances. This answers a question of Erdős, Hickerson and Pach. We then revisit an old problem of Erdős: given
any n points in the plane (or in d dimensions), how many of them can one select so that the distances which are determined are all distinct? — and provide (make
explicit) some new bounds in one and two dimensions. Other related distance problems are also discussed.