The Extremal Functions of Classes of Matroids of Bounded Branch-Width

Abstract

For a set of matroids M, let ex M (n) be the maximum size of a simple rank-n matroid in M. We prove that, for any finite field $$\mathbb{F}$$ , if M is a minor-closed class of $$\mathbb{F}$$ -representable matroids of bounded branch-width, then limn→ ∞ex M (n)/n exists and is a rational number, ∆. We also show that ex M (n) - ∆n is periodic when n is sufficiently large and that exM is achieved by a subclass of M of bounded path-width.