Sylvester matroid


In matroid theory, a Sylvester matroid is a matroid in which every pair of elements belongs to a three-element circuit of the matroid.

Example

The -point line is a Sylvester matroid because every pair of elements is a basis and every triple is a circuit.
A Sylvester matroid of rank three may be formed from any Steiner triple system, by defining the lines of the matroid to be the triples of the system. Sylvester matroids of rank three may also be formed from Sylvester–Gallai configurations, configurations of points and lines 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.

Properties

A Sylvester matroid with rank must have at least elements; this bound is tight only for the projective spaces over GF, of which the Fano plane is an example.
In a Sylvester matroid, every independent set can be augmented by one more element to form a circuit of the matroid.
Sylvester matroids cannot be represented over the real numbers, nor can they be oriented.

History

Sylvester matroids were studied and named by after James Joseph Sylvester, because they violate the Sylvester–Gallai theorem that for every finite set of points there is a line containing only two of the points.