We answer the following question posed by Paul Erdős and George Purdy: determine the largest number f d (k) = f with the property that almost all k-element subsets of any n-element set in R d determine at least f distinct distances, for all sufficiently large n. For d = 2 we investigate the asymptotic behaviour of the maximum number of k-element subsets of a set of n points, each subset determining at most i distinct distances, for some prespecified number i. We also show that if k = o(n 1 7 ), almost all k-element subsets of a planar point set determine distinct distances.
ASJC Scopus subject areas
- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics