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**Zentralblatt MATH**

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

**Zbl.No: ** 737.05006

**Autor: ** Erdös, Paul; Hickerson, Dean; Pach, János

**Title: ** A problem of Leo Moser about repeated distances on the sphere. (In English)

**Source: ** Am. Math. Mon. 96, No.7, 569-575 (1989).

**Review: ** We disprove a conjecture of Leo Moser by showing that (i) for every natural number n and 0 < \alpha < 2 there is a system of n points on the unit sphere S^{2} such that the number of pairs at distance \alpha from each other is at least \hbox{const}· n log^*n (where log^* stands for the iterated logarithm function) (ii) for every n there is a system of n points on S^{2} such that the number of pairs at distance \sqrt 2 from each other is at least \hbox{const}· n^{4/3}. We also construct a set of n points in the plane in general position (no 3 on a line, no 4 on a circle) such that they determine fewer than \hbox{const}· n^{ log 3/ log 2} distinct distances, which settles a problem of Erdös.

**Classif.: ** * 05A05 Combinatorial choice problems

05B30 Other designs, configurations

00A07 Problem books

**Keywords: ** conjecture of Leo Moser; problem of Erdös

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