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 and be compact, countable and disjoint sets of points.

(1) If spans (the Euclidean plane) then there must exist a line through two points of that intersects in only finitely many points.

(2) If spans (Euclidean three-space) then there must exist a line through exactly two points of .

(3) If spans then there must exist a line through at least two points of one of the sets that does not intersect the other set.

**[1]**J. M. Borwein,*Problem*297, Canad. Math. Bull.**23**(1980), 506.**[2]**P. B. Borwein,*Variations on Sylvester's Problem*, Proc. Atlantic Mathematics Days, Memorial University (1981), 39-43.**[3]**P. B. Borwein and M. Edelstein,*A conjecture related to Sylvester's Problem*, Amer. Math. Monthly**90**(1983), 389-390. MR**1540214****[4]**G. D. Chakerian,*Sylvester's problem on collinear points and a relative*, Amer. Math. Monthly**77**(1970), 164-167. MR**0258659 (41:3305)****[5]**E. Kamke,*Theory of sets*, Dover, New York, 1950. MR**0032709 (11:335a)****[6]**W. Moser,*Research problems in discrete geometry*, McGill Math. Rep. 81-3, McGill Univ., Montreal, Quebec.**[7]**T. S. Motzkin,*Nonmixed connecting lines*, Notices Amer. Math. Soc.**14**(1967), 837.**[8]**J. J. Sylvester,*Mathematical question 11851*, Educational Times**59**(1893), 98.**[9]**D. Tingley,*Topics related to Sylvester's Problem*, M. A. Thesis, Dalhousie Univ., Halifax, N.S., 1976.

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0733410-4

Article copyright:
© Copyright 1984
American Mathematical Society