Projective geometries in exponentially dense matroids. I

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