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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?)

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Article copyright: © Copyright 1984 American Mathematical Society