Sylvester-Gallai theorem and metric betweenness (Q1880209)

Sylvester conjectured in 1893 and T. Gallai proved in the early 1930s that every finite set \(S\) of points in the plane includes two points such that the line passing through them includes no other points of \(S\) or all other points of \(S\). This result will be referred to as the Sylvester-Gallai theorem. \textit{K. Menger} [Math.\ Ann.\ 100, 75--163 (1928; JFM 54.0622.02)] has introduced the notion of metric betweenness, which can be defined for any metric space \((X, \varrho)\) by stipulating that \(b\) lies between \(a\) and \(c\) if and only if \(\varrho(a,b)+\varrho(b,c)=\varrho(a,c)\). If we define a {line} in a metric space as the affine hull (defined by means of the betweenness notion induced by \(\varrho\)) of a set of two points, then the author conjectures that the Sylvester-Gallai theorem holds for all finite metric spaces \((S, \varrho)\). To support the conjecture, the author shows that it holds for metric spaces with at most 9 elements, and that it holds for all metric spaces induced by a finite connected graph. Other parts of the paper are devoted to a study of the axiomatizability of the metric betweenness relation, which the author, much like \textit{R. Mendris} and \textit{P. Zlatoš} [Proc. Am. Math. Soc. 123, No. 3, 873--882 (1995; Zbl 0818.03002)] (see also \textit{J. Šimko} [Proc.\ Am.\ Math.\ Soc.\ 127, No. 2, 323--325 (1999; Zbl 0906.03007)]), doubts ``can be described in a way that is both explicit and concise'', and to a characterization of \({\mathcal l}_1\)-betweenness. The remarkable fact is that the conjecture is true. It was proved by the author's student Xiaomin Chen. His proof is valid inside the axiom system proposed by \textit{M. Moszyńska} [Fundam.\ Math.\ 96, 17--29 (1977; Zbl 0355.50003)]. So far, the weakest axiom system in which the Sylvester-Gallai theorem was known to hold was that of ordered geometry (see \textit{H. S. M. Coxeter} [Introduction to geometry. 2nd ed. (Wiley, New York) (1969; Zbl 0181.48101)]). Moszyńska's axiom system is incomparable to (i.e. neither weaker nor stronger than) that of ordered geometry. For a similar situation see the reviewer's [Beitr.\ Algebra Geom.\ 42, No. 2, 401--406 (2001; Zbl 1008.52002)].