Growth rate functions of dense classes of representable matroids

TL;DR

This paper establishes precise upper bounds on the size of dense matroids within certain classes of $ ext{GF}(q^2)$-representable matroids, extending understanding of their growth rates and structural limitations.

Contribution

It provides tight bounds for the number of points in large rank matroids in specific minor-closed classes of $ ext{GF}(q^2)$-representable matroids, including constructions achieving these bounds.

Findings

Derived tight upper bounds for matroid sizes in classes containing all $ ext{GF}(q)$-representable matroids.

Constructed matroids that attain these bounds, demonstrating their optimality.

Applied results to bound the size of $ ext{GF}(q^2)$-representable matroids without certain minors.

Abstract

For each proper minor-closed subclass $\cM$ of the $\GF(q^2)$-representable matroids containing all simple $\GF(q)$-representable matroids, we give, for all large $r$, a tight upper bound on the number of points in a rank-$r$ matroid in $\cM$, and construct a rank-$r$ matroid in $\cM$ for which equality holds. As a consequence, we give a tight upper bound on the number of points in a $\GF(q^2)$-representable, rank-$r$ matroid with no $\PG(k,q^2)$-minor.