Abstract

The following three variations of Sylvester’s Problem are established. Let A A and B B be compact, countable and disjoint sets of points. (1) If A A spans E 2 {E^2} (the Euclidean plane) then there must exist a line through two points of A A that intersects A A in only finitely many points. (2) If A A spans E 3 {E^3} (Euclidean three-space) then there must exist a line through exactly two points of A A . (3) If A ∪ B A \cup B spans E 2 {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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