The extremal function for geometry minors of matroids over prime fields

Abstract

A frame template over a field $\mathbb{F}$ describes the precise way in which a given $\mathbb{F}$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is described by a given frame template over a prime field. Subject to the matroid minors hypothesis of Geelen, Gerards, and Whittle, we use our result to determine, for each projective or affine geometry $N$ over a prime field $\mathbb{F}$, a best-possible upper bound on the number of elements in a simple $\mathbb{F}$-representable matroid $M$ of sufficiently large rank with no $N$-minor.

References in corpus (1)

• Templates for Binary Matroids