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**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 C_{1}, ... ,C_{n} be n simple closed curves. Assume that C_{i} \cap C_{j} 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 x_{i}, i = 1, ... , \omega (n) so that to each i there exists j_{1} and j_{2} so that the only intersection of C_{j1} and C_{j2} is x_{i}. The authors prove: there exist constants c_{1},c_{2} > 0 such that c_{1}n^{4/3} < \omega (n) < c_{2}n^{5/3} and if the C_{i} are all circles there exists c_{3} such that \omega (n) > n^{1+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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