Projective geometries in exponentially dense matroids. II

We show for each positive integer a that, if M is a minor-closed class of matroids not containing all rank-(a+1) uniform matroids, then there exists an integer c such that either every rank-r matroid in M can be covered by at most rc rank-a sets, or M contains the GF(q)-representable matroids for some prime power q and every rank-r matroid in M can be covered by at most cqr rank-a sets. In the latter case, this determines the maximum density of matroids in M up to a constant factor.