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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
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?)

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