Exponentially Dense Matroids

This thesis deals with questions relating to the maximum density of rank-n matroids in a minor-closed class. Consider a minor-closed class M of matroids that does not contain a given rank-2 uniform matroid. The growth rate function is defined by hM(n) = max (|M | : M ∈M simple, r(M) ≤ n) . The Growth Rate Theorem, due to Geelen, Kabell, Kung, and Whittle, shows that the growth rate function is either linear, quadratic, or exponential in n. In the case of exponentially dense classes, we conjecture that, for sufficiently large n, hM(n) = q − 1 q − 1 − c, where q is a prime power, and k and c are non-negative integers depending only on M. We show that this holds for several interesting classes, including the class of all matroids with no U2,t-minor. We also consider more general minor-closed classes that exclude an arbitrary uniform matroid. Here the growth rate, as defined above, can be infinite. We define a more suitable notion of density, and prove a growth rate theorem for this more general notion, dividing minor-closed classes into those that are at most polynomially dense, and those that are exponentially dense.