## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  257.52016
Autor:  Erdös, Paul; Grünbaum, Branko
Title:  Osculation vertices in arrangements of curves. (In English)
Source:  Geometriae dedicata 1, 322-333 (1973); correction 3, 130 (1974).
Review:  Let C1, ... ,Cn be n simple closed curves. Assume that Ci \cap Cj is either empty or is a single point or is a pair of points at which the two curves cross each other. Denote by \omega (n) the largest integer for which there are n curves and \omega (n) points xi, i = 1, ... , \omega (n) so that to each i there exists j1 and j2 so that the only intersection of Cj1 and Cj2 is xi. The authors prove: there exist constants c1,c2 > 0 such that c1n4/3 < \omega (n) < c2n5/3 and if the Ci are all circles there exists c3 such that \omega (n) > n1+c3/ log log n. Several open related problems are discussed.
Classif.:  * 52A40 Geometric inequalities, etc. (convex geometry)
52C17 Packing and covering in n dimensions (discrete geometry)
05C99 Graph theory

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