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**Zentralblatt MATH**

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

**Zbl.No: ** 831.52008

**Autor: ** Erdös, Paul; Fishburn, Peter C.

**Title: ** Multiplicities of interpoint distances in finite planar sets. (In English)

**Source: ** Discrete Appl. Math. 60, No.1-3, 141-147 (1995).

**Review: ** For a set X of n points in the plane, let d_{1}, ..., d_{m} denote the different positive distances between the points of X, and r_{k} the multiplicity of d_{k}. The authors study the vector r(X) = (r_{1}, ..., r_{m}), where the numbering is chosen such that r_{1} \geq r_{2} \geq ··· \geq r_{m}. The case where X is the set V of vertices of a convex polygon is considered particularly. For n = 5 and m in **{**2,3**}**, the possible vectors r(X) and r(V) are completely specified. For n = 6, it is shown that r(X) cannot be equal to (7,7,1). There is a discussion of some known results and several challenging conjectures which are related to this topic.

**Reviewer: ** J.Linhart (Salzburg)

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

**Keywords: ** minimum number of different distances; multiplicity vector

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