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Sylvester–Gallai configuration
In geometry, a Sylvester–Gallai configuration consists of a finite subset of the points of a projective space with the property that the line through any two of the points in the subset also passes through at least one other point of the subset.
Instead of defining Sylvester–Gallai configurations as subsets of the points of a projective space, they may be defined as abstract incidence structures of points and lines, satisfying the properties that, for every pair of points, the structure includes exactly one line containing the pair and that every line contains at least three points. In this more general form they are also called Sylvester–Gallai designs. A closely related concept is a Sylvester matroid, a matroid with the same property as a Sylvester–Gallai configuration of having no two-point lines.
==Real and complex embeddability==
In the Euclidean plane, the real projective plane, higher-dimensional Euclidean spaces or real projective spaces, or spaces with coordinates in an ordered field, the Sylvester–Gallai theorem shows that the only possible Sylvester–Gallai configurations are one-dimensional: they consist of three or more collinear points.
was inspired by this fact and by the example of the Hesse configuration to ask whether, in spaces with complex-number coordinates, every Sylvester–Gallai configuration is at most two-dimensional. repeated the question. answered Serre's question affirmatively; simplified Kelly's proof, and proved analogously that in spaces with quaternion coordinates, all Sylvester–Gallai configurations must lie within a three-dimensional subspace.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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