The -point line (i.e., the rank 2 uniform matroid on elements, ) 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 (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 configuration.
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.
- Murty, U. S. R. (1969), "Sylvester matroids", Recent Progress in Combinatorics (Proc. Third Waterloo Conf. on Combinatorics, 1968), New York: Academic Press, pp. 283–286, MR 0255432.
- Welsh, D. J. A. (2010), Matroid Theory, Courier Dover Publications, p. 297, ISBN 9780486474397.
- Murty, U. S. R. (1970), Matroids with Sylvester property, Aequationes Mathematicae 4: 44–50, doi:10.1007/BF01817744, MR 0265186.
- Bryant, V. W.; Dawson, J. E.; Perfect, Hazel (1978), Hereditary circuit spaces, Compositio Mathematica 37 (3): 339–351, MR 511749.
- Ziegler, Günter M. (1991), Some minimal non-orientable matroids of rank three, Geometriae Dedicata 38 (3): 365–371, doi:10.1007/BF00181199, MR 1112674.