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 tn, n = 2,3,..., be the number of lines of the arrangement containing exactly n points; and let \epsilon be the lesser of {t3/ t2, 1}. It is shown that absolute positive constants C1, C2 exist such that if the number of points is n, the total number of lines determined by the points is at least C1 in n2; and t3 is at least C2 \epsilon2 n2.
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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