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Sylvester's problem and Motzkin's theorem for countable and compact sets


Author: Peter B. Borwein
Journal: Proc. Amer. Math. Soc. 90 (1984), 580-584
MSC: Primary 52A37
DOI: https://doi.org/10.1090/S0002-9939-1984-0733410-4
MathSciNet review: 733410
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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 [Enhancements On Off] (What's this?)

  • [1] J. M. Borwein, Problem 297, Canad. Math. Bull. 23 (1980), 506.
  • [2] P. B. Borwein, Variations on Sylvester's Problem, Proc. Atlantic Mathematics Days, Memorial University (1981), 39-43.
  • [3] 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, https://doi.org/10.2307/2975576
  • [4] G. D. Chakerian, Sylvester’s problem on collinear points and a relative, Amer. Math. Monthly 77 (1970), 164–167. MR 0258659, https://doi.org/10.2307/2317330
  • [5] E. Kamke, Theory of Sets. Translated by Frederick Bagemihl, Dover Publications, Inc., New York, N. Y., 1950. MR 0032709
  • [6] W. Moser, Research problems in discrete geometry, McGill Math. Rep. 81-3, McGill Univ., Montreal, Quebec.
  • [7] T. S. Motzkin, Nonmixed connecting lines, Notices Amer. Math. Soc. 14 (1967), 837.
  • [8] J. J. Sylvester, Mathematical question 11851, Educational Times 59 (1893), 98.
  • [9] D. Tingley, Topics related to Sylvester's Problem, M. A. Thesis, Dalhousie Univ., Halifax, N.S., 1976.

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DOI: https://doi.org/10.1090/S0002-9939-1984-0733410-4
Article copyright: © Copyright 1984 American Mathematical Society