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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 798.52017

**Autor: ** Erdös, Paul; Makai, Endre; Pach, János

**Title: ** Nearly equal distances in the plane. (In English)

**Source: ** Comb. Probab. Comput. 2, No.4, 401-408 (1993).

**Review: ** The authors prove that for every positive integer k and for every \epsilon > 0 there exist numbers n_{0} > 0 and c > 0 such that every set of n > n_{0} points in the Euclidean plane in pairwise distances at least 1 has the following property: for arbitrary reals t_{1},..., t_{k}, the number of pairs of points whose distance belongs to the set \bigcup_{i = 1}^{k} [t_{i}, t_{i}+c\sqrt{n}] is at most (n^{2}/2) (1- 1/(k+1)+\epsilon). This bound is asymptotically best possible. The proof generalizes the considerations of the authors and *J. Spencer* [DIMACS, Ser. Discret. Math. Theor. Comput. Sci. 4, 265-273 (1992; Zbl 741.52010)].

**Reviewer: ** M.Lassak (Bydgoszcz)

**Classif.: ** * 52C10 Erdoes problems and related topics of discrete geometry

**Keywords: ** distance; graph; subgraph; points; Euclidean plane

**Citations: ** Zbl 741.52010

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