Quadratically Dense Matroids

This thesis is concerned with finding the maximum density of rank-$n$ matroids in a minor-closed class. The extremal function of a non-empty minor-closed class $\mathcal M$ of matroids which excludes a rank-2 uniform matroid is defined by $$h_{\mathcal M}(n)=\max(|M|\colon M\in \mathcal M \text{ is simple, and } r(M)\le n).$$ The Growth Rate Theorem of Geelen, Kabell, Kung, and Whittle shows that this function is either linear, quadratic, or exponential in $n$. In this thesis we prove a general result about classes with quadratic extremal function, and then use it to determine the extremal function for several interesting classes of representable matroids, for sufficiently large integers $n$. In particular, for each integer $t\ge 4$ we find the extremal function for all but finitely many $n$ for the class of $\mathbb C$-representable matroids with no $U_{2,t}$-minor, and we find the extremal function for the class of matroids representable over finite fields $\mathbb F_1$ and $\mathbb F_2$ where $|\mathbb F_1|-1$ divides $|\mathbb F_2|-1$ and $|\mathbb F_1|$ and $|\mathbb F_2|$ are relatively prime