  We show that, if $M$ is a simple rank-$n$ matroid with no $\ell$-point line
minor and no minor isomorphic to the cycle matroid of a $t$-vertex complete
graph, then the ratio $|M| / n$ is bounded above by a singly exponential
function of $\ell$ and $t$. We also bound this ratio in the special case where
$M$ is a frame matroid, obtaining an answer that is within a factor of two of
best-possible.
