Sylvester matroid

It is a Sylvester matroid because every pair of elements is a basis and every triple is a circuit.

Sylvester matroids of rank three may also be formed from Sylvester–Gallai configurations, configurations of points and lines (in non-Euclidean spaces) with no two-point line.

For example, the Fano plane and the Hesse configuration give rise to Sylvester matroids with seven and nine elements respectively, and may be interpreted either as Steiner triple systems or as Sylvester–Gallai configurations.

elements; this bound is tight only for the projective spaces over GF(2), of which the Fano plane is an example.

) cannot be represented over the real numbers (this is the Sylvester–Gallai theorem), nor can they be oriented.

[5] Sylvester matroids were studied and named by Murty (1969) after James Joseph Sylvester, because they violate the Sylvester–Gallai theorem (for points and lines in the Euclidean plane, or in higher-dimensional Euclidean spaces) that for every finite set of points there is a line containing only two of the points.