The extremal function for geometry minors of matroids over prime fields

TL;DR

This paper characterizes the maximum size of large, simple, prime-field representable matroids that exclude a fixed projective or affine geometry minor, using frame templates and the matroid minors hypothesis.

Contribution

It determines the extremal function for geometry minors of matroids over prime fields via frame templates and the matroid minors hypothesis.

Findings

Identifies the maximum rank projective or affine geometry described by a frame template.

Provides upper bounds on the size of matroids excluding a fixed geometry minor.

Connects the extremal problem to the matroid minors hypothesis.

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.