2013/02/21 by William Gasarch, Gasarch, William, Sam Zbarsky +1
Computer Science · Mathematics · #05D10 #52C10 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #math.CO #msc:05D10 #msc:52C10
paper · pdf · doi:10.48550/arxiv.1302.5334
22 pages
arxiv created 2013/02/21 · openalex publication_date 2013/02/21 · arxiv updated 2013/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let P be a set of n points in Rd. How big is the largest subset X of P such that all of the distances determined between pairs are different? We show that X is at at least Omega(n1/6d) This is not the best known; however the technique is new. Assume that no three of the original points are collinear. How big is the largest subset X of P such that all of the areas determined by elements of all triples are different? We show that, if d=2 then X is at least Omega((log log n)1/186) and if d=3 then X is at least Omega((log log n)1/396). We also obtain results for countable sets of points in Rd. All of our proofs use variants of the canonical Ramsey theorem and some geometric lemmas.