Clique minors in dense matroids

The objective of this thesis is to bound the number of points a $U_{2,\ell+2}$- and $M(K_{k+1})$-minor-free matroid has. We first prove that a sufficiently large matroid will contain a structure called a tower. We then use towers to find a complete minor in a matroid with no $U_{2,\ell+2}$-minor

On the density of matroids omitting a complete-graphic minor

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.Comment: 25 page