  We show that, if $k$ and $\ell$ are positive integers and $r$ is sufficiently
large, then the number of rank-$k$ flats in a rank-$r$ matroid $M$ with no
$U_{2,\ell+2}$-minor is less than or equal to number of rank-$k$ flats in a
rank-$r$ projective geometry over GF$(q)$, where $q$ is the largest prime power
not exceeding $\ell$.
