Projective geometries in dense matroids

We prove that, given integers l , q ⩾ 2 and n there exists an integer α such that, if M is a simple matroid with no l + 2 point line minor and at least α q r ( M ) elements, then M contains a PG ( n − 1 , q ′ ) -minor, for some prime-power q ′ > q .