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 S2 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 S2 such that the number of pairs at distance \sqrt 2 from each other is at least \hbox{const}· n4/3. 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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