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**Zentralblatt MATH**

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

**Zbl.No: ** 345.52007

**Autor: ** Erdös, Paul; Purdy, George

**Title: ** Some extremal problems in geometry. IV. (In English)

**Source: ** Proc. 7th southeast. Conf. Comb., Graph Theory, Comput.; Baton Rouge 1976, 307-322 (1976).

**Review: ** [For the entire collection see Zbl 328.00003.]

The authors discuss some questions and obtain new results on bounds for several functions occurring in problems of Combinatorial Geometry, mostly in the plane. Examples: Let f(n) denote the maximum number of times that unit distance can occur among n points in the plane if no three points lie on a line; then f(n) \geq 2n log ^{n}/_{6} /3 log 3. Let g(n) be the minimum number of triangles of different areas which must occur among n points in the plane, not all on a line; then c_{1}n^{3/4} \leq g(n) \leq c_{2}n. Let f(n) be the minimum number k such that there exist k points in the n by n lattice L_{n} so that the lines through any two of them cover all the points of L_{n}; then f(n) \geq cn^{2/3}. Other similar problems are concerned with congruent triangles, isosceles triangles, congruent or incongruent subsets, always taken from n given points.

**Reviewer: ** R.Schneider

**Classif.: ** * 52A40 Geometric inequalities, etc. (convex geometry)

51M25 Length, area and volume (geometry)

00A07 Problem books

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