# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Sylvester’s problem and Motzkin’s theorem for countable and compact setsHTML articles powered by AMS MathViewer

by Peter B. Borwein
Proc. Amer. Math. Soc. 90 (1984), 580-584 Request permission

## Abstract:

The following three variations of Sylvester’s Problem are established. Let \$A\$ and \$B\$ be compact, countable and disjoint sets of points. (1) If \$A\$ spans \${E^2}\$ (the Euclidean plane) then there must exist a line through two points of \$A\$ that intersects \$A\$ in only finitely many points. (2) If \$A\$ spans \${E^3}\$ (Euclidean three-space) then there must exist a line through exactly two points of \$A\$. (3) If \$A \cup B\$ spans \${E^2}\$ then there must exist a line through at least two points of one of the sets that does not intersect the other set.
References
J. M. Borwein, Problem 297, Canad. Math. Bull. 23 (1980), 506. P. B. Borwein, Variations on Sylvester’s Problem, Proc. Atlantic Mathematics Days, Memorial University (1981), 39-43.
• Peter Borwein and Michael Edelstein, Unsolved Problems: A Conjecture Related to Sylvester’s Problem, Amer. Math. Monthly 90 (1983), no. 6, 389–390. MR 1540214, DOI 10.2307/2975576
• G. D. Chakerian, Sylvester’s problem on collinear points and a relative, Amer. Math. Monthly 77 (1970), 164–167. MR 258659, DOI 10.2307/2317330
• E. Kamke, Theory of Sets. Translated by Frederick Bagemihl, Dover Publications, Inc., New York, N.Y., 1950. MR 0032709
• W. Moser, Research problems in discrete geometry, McGill Math. Rep. 81-3, McGill Univ., Montreal, Quebec. T. S. Motzkin, Nonmixed connecting lines, Notices Amer. Math. Soc. 14 (1967), 837. J. J. Sylvester, Mathematical question 11851, Educational Times 59 (1893), 98. D. Tingley, Topics related to Sylvester’s Problem, M. A. Thesis, Dalhousie Univ., Halifax, N.S., 1976.
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