Simple matroids with bounded cocircuit size (Q2709845)

The simple observation that a graph on \(2n\) vertices, each of which has degree at most \(d\), can have at most \(nd\) edges is generalized in the paper under review to matroids, where \(d\) becomes the cocircuit size and the role of the vertices is replaced by the rank of the matroid. It is shown that a simple rank-3 matroid with cocircuit size bounded by \(d\) has at most \(d+d^{1/2} +1\) points. Rank-3 geometries whose number of points equals the floor of this bound are classified. They comprise three infinite families arising from affine and projective planes as well as some special examples for small \(d\), e.g. for \(d=8\) the unique extremal example is the Nwankpa plane. Bounds for the number of points of rank-4 and rank-5 geometries with bounded cocircuit size are obtained under certain connectivity assumptions.