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On a Problem of Groübaum

Published online by Cambridge University Press:  20 November 2018

P. Erdös
P. Erdös*
Affiliation:
University of Waterloo, Waterloo, OntarioUniversity of Alberta, Edmonton, Alberta
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P n will denote a set of n points in the plane. A well known theorem of Gallai- Sylvester (see e.g. [4]) states that if the points of P n do not all lie on a line then they always determine an ordinary line, i.e. a line which goes through precisely two of the points of P n .

Using this theorem I proved that if the points do not all lie on a line, they determine at least n lines. I conjectured that if n>n0 and no n—1 points of Pn are on a line, they determine at least 2n-4 lines. This conjecture was proved by Kelly and Moser [3], who, in fact, proved the following more general result: Let P n be such that at most n—k of its points are collinear.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972, pp. 23 - 25
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Elliott, P. D. T. A., On the number of circles determined by n points, Acta. Math. Acad Sci. Hungar. 10 (1967), 181-188.Google Scholar
2. As far as known to the author this result was first proved by Hanani in 1938 (he published his proof only later) and it was first published in de Bruijn, N. G. and Erdös, P., On a combinatorial problem, Indig. Math. 10 (1948), 421-423.Google Scholar
3. Kelly, L. M. and Moser, W., On the number of ordinary lines determined by n points, Canad. J. Math. 10 (1958), 210-219.Google Scholar
4. Motzkin, Th., The lines and planes connecting the points of a finite set, Trans. Amer. Math. Soc. 70 (1951), 451-464.Google Scholar