On The Density of Binary Matroids Without a Given Minor

Abstract

This thesis is motivated by the following question: how many elements can a simple binary matroid with no $\PG(t,2)$-minor have? This is a natural analogue of questions asked about the density of graphs in minor-closed classes. We will answer this question by finding the eventual growth rate function of the class of matroids with no $\PG(t,2)$-minor, for any $t\ge 2$. Our main tool will be the matroid minors structure theory of Geelen, Gerards, and Whittle, and much of this thesis will be devoted to frame templates, the notion of structure in that theory.