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 lim_{n→ ∞}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 ex_M is achieved by a subclass of M of bounded path-width.