Point sets with many unit circles. (English) Zbl 0638.52005

Let f(n) denote the largest integer for which there are n points in the Euclidean plane with the property that there are f(n) distinct circles with the same radii passing through at least 3 of the n points. The authors establish \(f(3)=1\), \(f(4)=f(5)=4\), \(f(6)=8\), \(f(7)=12\), and announce that \(f(8)=16\).
Reviewer: A.Cohen


52A37 Other problems of combinatorial convexity
52A40 Inequalities and extremum problems involving convexity in convex geometry
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[1] Elekes, G., \(n\) points in the plane can determine \(n^{32}\) unit circles, Combinatorica, 4, 131 (1984) · Zbl 0561.52009
[2] Erdös, P., On some of my conjectures in Number Theory and Combinatorics, Congressus Numerantium, 39, 3-20 (1983)
[3] Harborth, H., Einheitskreise in ebenen Punktmengen, (3. Kolloquium über Diskrete Geometrie (1985), Institut für Mathematik der Universität Salzburg), 163-168 · Zbl 0572.52020
[4] Moser, W.; Pach, J., Research Problems in Discrete Geometry (1984), problem 25.
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