We show for each positive integer a that, if \cM is a minor-closed class
of matroids not containing all rank-(a+1) uniform matroids, then there exists
an integer n such that either every rank-r matroid in \cM can be covered
by at most rn sets of rank at most a, or \cM contains the
\GF(q)-representable matroids for some prime power q, and every rank-r
matroid in \cM can be covered by at most rnqr sets of rank at most a.
This determines the maximum density of the matroids in \cM up to a polynomial
factor