The Sylvester–Gallai theorem
A finite set of points in the plane does not lie on a single line. Prove that some line passes through exactly two of the points.
History and references
Solved in 1941 by Eberhard Melchior (1941) and Tibor Gallai (1944).
Sylvester posed the problem in 1893, and a proof published that year by H. J. Woodall was already noted to be incomplete. Eberhard Melchior proved it in 1941 in a dual form about arrangements of lines, and Paul Erdős, unaware of this, raised it again in 1943, after which Tibor Gallai gave a proof. A well-known short proof by Leroy Milton Kelly takes, among all pairs of a point and a connecting line not through it, the pair at the smallest positive distance, and shows by contradiction that this line contains only two points. The statement fails over the complex numbers, where the nine inflection points of a cubic curve form a configuration in which every line through two of them contains a third. Stronger versions show that there are always at least linearly many such lines.
References
Solutions 0
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