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**Zentralblatt MATH**

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

**Zbl.No: ** 831.52009

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

**Title: ** Intervertex distances in convex polygons. (In English)

**Source: ** Discrete Appl. Math. 60, No.1-3, 149-158 (1995).

**Review: ** Let V be the set of vertices of a convex n-gon in the plane. Denote by d_{1}, ..., d_{m} the different positive distances between the points of V, and by r_{k} the multiplicity of d_{k}. Choose the numbering such that r_{1} \geq r_{2} \geq ··· \geq r_{m}. For fixed n, the maximum of r_{i} over all convex n-gons is denoted by r_{i} (n). The values of r_{1} (n) and r_{2} (n) are known for n \leq 8. In particular we have r_{2} (n) \leq n in this case. Here a construction is presented which shows r_{2} (25) \geq 26 and \sup_{n} r_{2} (n)/n \geq 7/6.

A monotone sequence in V from v_{0} is a sequence of vertices v_{0}, v_{1}, ..., v_{k} in which the v_{i} are encountered in succession going (counter-)clockwise from v_{0}, such that the distance from v_{0} to v_{i} is strictly increasing. Let g(n) denote the minimum (over all convex n-gons) of the maximum length of monotone sequences. In a previous paper, the authors have shown \lfloor n/3 \rfloor+1 \leq g(n). Here, g(n) \leq \lceil n/3 \rceil+2 is proved.

**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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