Sylvester's Line Problem

Sylvester's line problem, known as the Sylvester-Gallai theorem in proved form, states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single line. This problem was proposed by Sylvester (1893), who asked readers to "Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line."

Woodall (1893) published a four-line "solution," but an editorial comment following his result pointed out two holes in the argument and sketched another line of enquiry, which is characterized as "equally incomplete, but may be worth notice." However, no correct proof was published at the time (Croft et al. 1991, p. 159), but the problem was revived by Erdős (1943) and correctly solved by Grünwald (1944). Coxeter (1948, 1969) transformed the problem into an elementary form, and a very short proof using the notion of Euclidean distance was given by Kelly (Coxeter 1948, 1969; Chvátal 2004). The theorem also follows using projective duality from a result of Melchior (1940) proved by a simple application of Euler's polyhedral formula (Chvátal 2004).

Additional information on the theorem can be found in Borwein and Moser (1990), Erdős and Purdy (1991), Pach and Agarwal (1995), and Chvátal (2003).

In September 2003, X. Chen proved a conjecture of Chvátal that, with a certain definition of a line, the Sylvester-Gallai theorem extends to arbitrary finite metric spaces.

See also

Collinear, de Bruijn-Erdős Theorem, Orchard-Planting Problem, Sylvester's Four-Point Problem

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Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 17-19, 2003.Borwein, P. and Moser, W. O. J. "A Survey of Sylvester's Problem and Its Generalizations." Aeq. Math. 40, 111-135, 1990.Chen, X. "The Sylvester-Chvátal Theorem." Preprint. Nov. 4, 2003.átal, V. "Sylvester-Gallai Theorem and Metric Betweenness." Disc. Comput. Geom. 31, 175-195, 2004.Coxeter, H. S. M. "A Problem of Collinear Points." Amer. Math. Monthly 55, 26-28, 1948.Coxeter, H. S. M. §4.7 and 12.3 in Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 159, 1991.Erdős, P. "Problem 4065: Three Point Collinearity." Amer. Math. Monthly 50, 65, 1943.Erdős, P. and Purdy, G. "Some Extremal Problems in Combinatorial Geometry." In Handbook of Combinatorics, Vol. 1 (Ed. R. L. Graham, M. Grötschel, and L. Lovász). Amsterdam, Netherlands: North-Holland, pp. 809-874, 1991.Grünwald, T. "Solution to Problem 4065." Amer. Math. Monthly 51, 169-171, 1944.Hansen, S. Contributions to the Sylvester-Gallai Theory. Ph.D. thesis. Copenhagen, Denmark: University of Copenhagen, 1981.Kelly, L. M. and Moser, W. O. J. "On the Number of Ordinary Lines Determined by n Points." Canad. J. Math. 10, 210-219, 1958.Melchior, E. "Über Vielseite der Projektive Ebene." Deutsche Math. 5, 461-475, 1940.Pach, J. and Agarwal, P. K. Combinatorial Geometry. New York: Wiley, 1995.Sylvester, J. J. Educational Times, 46, No. 383, 156, March 1, 1893.Woodall, H. J. Item 11851. Educational Times 46, No. 385, p. 231, May 1, 1893.

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Sylvester's Line Problem

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Weisstein, Eric W. "Sylvester's Line Problem." From MathWorld--A Wolfram Web Resource.

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