Sylvester’s problem and Motzkin’s theorem for countable and compact sets
Author:
Peter B. Borwein
Journal:
Proc. Amer. Math. Soc. 90 (1984), 580584
MSC:
Primary 52A37
DOI:
https://doi.org/10.1090/S00029939198407334104
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 threespace) 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.

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Article copyright:
© Copyright 1984
American Mathematical Society