2008/04/08 by Sascha Kurz, Kurz, Sascha, Alfred Wassermann +1 · 1 citation
Mathematics · #11D99 #52C10 (Primary) #53C65 (Secondary) #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications #math.CO #msc:11D99 #msc:52C10 #msc:53C65
paper · pdf · doi:10.48550/arxiv.0804.1307
12 pages, 5 figures
arxiv created 2008/04/08 · openalex publication_date 2008/04/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2,n) of a plane integral point set consisting of n points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2,n) achieving the known upper bound nc2loglog n up to a constant in the exponent. A famous question of Erdős asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by d(2,n). Recently d(2,7)=22 270 could be determined via an exhaustive search.