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**Zentralblatt MATH**

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

**Zbl.No: ** 826.52009

**Autor: ** Erdös, Paul; Purdy, G.

**Title: ** Two combinatorial problems in the plane. (In English)

**Source: ** Discrete Comput. Geom. 13, No.3-4, 441-443 (1995).

**Review: ** This paper contains the authors' solution to one problem about arrangements of lines and points in the plane, and a partial solution, due to Dean Hickerson, of another. (The connecting theme is that both problems were posed in a 1978 paper by the same authors.) Let t_{n}, n = 2,3,..., be the number of lines of the arrangement containing exactly n points; and let \epsilon be the lesser of **{**t_{3}/ t_{2}, 1**}**. It is shown that absolute positive constants C_{1}, C_{2} exist such that if the number of points is n, the total number of lines determined by the points is at least C_{1} in n^{2}; and t_{3} is at least C_{2} \epsilon^{2} n^{2}.

The second problem asks how small a set T can be, if there is an n-point noncollinear set S, disjoint from T, such that every line through two or more points of S contains a point of T. For n \geq 6, a construction, due to Hickerson, is given for a pair (S,T) such that |S| = n, |T| = n-2.

**Reviewer: ** R.Dawson (Halifax)

**Classif.: ** * 52A37 Other problems of combinatorial convexity

00A07 Problem books

**Keywords: ** arrangements; lines; points; plane

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