Growth rate functions of dense classes of representable matroids

For each proper minor-closed subclassM of the GF(q)representable matroids containing all GF(q)-representable matroids, we give, for all large r, a tight upper bound on the number of points in a rank-r matroid inM, and give a rank-r matroid inM for which equality holds. As a consequence, we give a tight upper bound on the number of points in a GF(q)-representable, rank-r matroid of large rank with no PG(k, q)-minor.