Nearly equal distances and Szemerédi's regularity lemma
نویسندگان
چکیده
منابع مشابه
Nearly Equal Distances in the Plane
distances determined by them? In particular, what is the maximum number of pairs of points that determine the same distance? Although a lot of progress has been made in this area, we are still very far from having satisfactory answers to the above questions (cf. [EP], [MP], [PA] for recent surveys). Two distances are said to be nearly the same if they differ by at most 1. If all points of a set...
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Let (X, d) be any finite metric space with n elements. We show that there are two pairs of distinct elements in X that determine two nearly equal distances in the sense that their ratio differs from 1 by at most 9 logn n2 . This bound (apart for the multiplicative constant) is best possible and we construct a metric space that attains this bound. We discus related questions and consider in part...
متن کاملSzemerédi’s Regularity Lemma
Szemerédi’s Regularity Lemma is an important result in extremal graph theory. Roughly speaking, the lemma states that every graph can be approximated by random graphs; that is, the vertex set of every graph can be split into equal size subsets such that the distribution of the edges between almost any two of these subsets is pseudorandom. The Regularity Lemma has already proved to be a powerful...
متن کاملSzemerédi Regularity Lemma
Szemerédi’s Regularity Lemma is one of the few truly universal tools in modern combinatorics, with numerous important applications. In particular, this lemma is the cornerstone of the theory of convergent sequences of dense graphs launched recently by Lovász and Szegedy [15], Borgs, Chayes, Lovász, Sós and Vesztergombi [3], [4] and Borgs, Chayes and Lovász [5]. The germ of a similar theory for ...
متن کاملAn Abstract Regularity Lemma
We extend in a natural way Szemerédis Regularity Lemma to abstract measure spaces. 1 Introduction In this note we extend Szemerédis Regularity Lemma (SRL) to abstract measure spaces. Our main aim is to nd general conditions under which the original proof of Szemerédi still works. Another extension of SRL to probality spaces was proved by Tao [3], but his results do not imply our most general...
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ژورنال
عنوان ژورنال: Computational Geometry
سال: 2006
ISSN: 0925-7721
DOI: 10.1016/j.comgeo.2005.06.002