Linearly-dense classes of matroids with bounded branch-width

Abstract

Let $M$ be a non-empty minor-closed class of matroids with bounded branch-width that does not contain arbitrarily large simple rank-$2$ matroids. For each non-negative integer $n$ we denote by $ex(n)$ the size of the largest simple matroid in $M$ that has rank at most $n$. We prove that there exist a rational number $\Delta$ and a periodic sequence $(a_0,a_1,\ldots)$ of rational numbers such that $ex(n) = \Delta n+a_n$ for each sufficiently large integer $n$.