Sylvester–Gallai theorem
The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.
A line that contains exactly two of a set of points is known as an ordinary line. Another way of stating the theorem is that every finite set of points that is not collinear has an ordinary line. According to a strengthening of the theorem, every finite point set (not all on one line) has at least a linear number of ordinary lines. An algorithm can find an ordinary line in a set of points in time .
History[edit]
The Sylvester–Gallai theorem was posed as a problem by J. J. Sylvester (1893). Kelly (1986) suggests that Sylvester may have been motivated by a related phenomenon in algebraic geometry, in which the inflection points of a cubic curve in the complex projective plane form a configuration of nine points and twelve lines (the Hesse configuration) in which each line determined by two of the points contains a third point. The Sylvester–Gallai theorem implies that it is impossible for all nine of these points to have real coordinates.[1]
H. J. Woodall (1893a, 1893b) claimed to have a short proof of the Sylvester–Gallai theorem, but it was already noted to be incomplete at the time of publication. Eberhard Melchior (1941) proved the theorem (and actually a slightly stronger result) in an equivalent formulation, its projective dual. Unaware of Melchior's proof,[2] Paul Erdős (1943) again stated the conjecture, which was subsequently proved by Tibor Gallai, and soon afterwards by other authors.[3]
In a 1951 review, Erdős called the result "Gallai's theorem",[4] but it was already called the Sylvester–Gallai theorem in a 1954 review by Leonard Blumenthal.[5] It is one of many mathematical topics named after Sylvester.