Matroids Denser than a Projective Geometry

The growth-rate function for a minor-closed class M of matroids is the function h where, for each non-negative integer r, h(r) is the maximum number of elements of a simple matroid in M with rank at most r. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential with some prime power q as the base, or infinite. Morover, if the growthrate function is exponential with base q, then the class contains all GF(q)-representable matroids, and so h(r) ≥ q −1 q−1 for each r. We characterise the classes that satisfy h(r) = q −1 q−1 for all sufficiently large r. As a consequence, we determine the eventual value of the growth rate function for most classes defined by excluding lines, free spikes and/or free swirls.