Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T07:20:44.588Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Problem of Groübaum

Published online by Cambridge University Press:  20 November 2018

P. Erdös
P. Erdös*
Affiliation:
University of Waterloo, Waterloo, Ontario University of Alberta, Edmonton, Alberta
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Pn 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 Pn 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 Pn.

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 Pn be such that at most n—k of its points are collinear.

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