Nearly equal distances and Szemerédi's regularity lemma

Authors:
János Pach;Radoš Radoičić;Jan Vondrák
Affiliations:
City College, CUNY and Courant Institute of Mathematical Sciences, New York University, New York, NY;Department of Mathematics, Rutgers University, Piscataway, NJ;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA
Venue:
Computational Geometry: Theory and Applications - Special issue on the Japan conference on discrete and computational geometry 2004
Year:
2006

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Abstract

A point set is separated if the minimum distance between its elements is one. Two numbers are called nearly equal if they differ by at most one. If a fixed positive percentage of all pairs of points belonging to a separated set of size n in R3 determine nearly equal distances, then the diameter of the set is at least constant times n. This proves a conjecture of Erdös.