A general approach to d-dimensional geometric queries
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Partitioning complete bipartite graphs by monochromatic cycles
Journal of Combinatorial Theory Series B
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
Extremal Graph Theory
Crossing patterns of semi-algebraic sets
Journal of Combinatorial Theory Series A
Hi-index | 0.00 |
A point set is separated if the minimum distance between its elements is 1. We call two real numbers nearly equal if they differ by at most 1. We prove that for any dimension d ≥ 2 and any γ 0, if P is a separated set of n points in Rd such that at least γn2 pairs in (P 2) determine nearly equal distances, then the diameter of P is at least C(d, γ)n2/(d-1) for some constant C(d, γ) 0. In the case of d = 3, this result confirms a conjecture of Erdös. The order of magnitude of the above bound cannot be improved for any d.