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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 E1 (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 E3 (Euclidean three-space) then there must exist a line through exactly two points of A. (3) If A U B spans E2 then there must exist a line through at least two points of one of the sets that does not intersect the other set.doi:10.1090/s0002-9939-1984-0733410-4 fatcat:qn24qcy4xnhobhcxq3w4q64pua