🔖 Sylvester’s Line Problem | Wolfram MathWorld

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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 ErdŇĎs (1943) and correctly solved by Gr√ľnwald (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; Chv√°tal 2004). The theorem also follows using projective duality from a result of Melchior (1940) proved by a simple application of Euler's¬†polyhedral formula¬†(Chv√°tal 2004).

Additional information on the theorem can be found in Borwein and Moser (1990), ErdŇĎs and Purdy (1991), Pach and Agarwal (1995), and Chv√°tal (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.

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