Sylvester-Gallai Theorem and Metric Betweenness

Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the Sylvester-Gallai theorem generalizes as follows: in every finite metric space, there is a line consisting of either two points or all the points of the space. Then we present meagre evidence in support of this rash conjecture and finally we discuss the underlying ternary relation of metric betweenness. 1 The Sylvester-Gallai theorem In March 1893, Sylvester [33] proposed the following problem: 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. The May 1893 issue of the same journal reports a four-line “solution” proposed by H.J. Woodall, A.R.C.S., followed by a comment pointing out two flaws in the argument and sketching another line of enquiry, which “. . . is equally incomplete, but may be worth notice”. Some forty years later, Erdös revived the problem and the first prooof was found shortly afterwards by T. Gallai (named Grünwald at that time): see Erdös [15]. Additional proofs were given by R.C. Buck, N.E. Steenrod, and R. Steinberg; in particular, Steinberg’s proof may be seen as a projective variation on Gallai’s affine theme; Coxeter ([10]; §12.3 of [11]) transformed it into an even more elementary form. The theorem also follows, through projective duality, from a result of Melchior [24], proved by a simple application of Euler’s polyhedral formula. A very short proof was given by L.M. Kelly; this proof uses the notion of Euclidean distance and can be found in Coxeter ([10]; §4.7 of [11]) and Gale (Chapter 8 of [18]). Further information