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Sylvester–Gallai Theorem and Metric Betweenness

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Sylvester conjectured in 1893 and Gallai proved some 40 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.

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Correspondence to Vasek Chvátal.

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Chvátal, V. Sylvester–Gallai Theorem and Metric Betweenness. Discrete Comput Geom 31, 175–195 (2004). https://doi.org/10.1007/s00454-003-0795-6

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  • Euclidean Space
  • Ternary Relation
  • Gallai Theorem