On minor-closed classes of matroids with exponential growth rate

TL;DR

This paper characterizes the growth rates of minor-closed classes of matroids with exponential density, showing exact formulas for the maximum size of matroids in classes excluding certain lines.

Contribution

It proves precise formulas for the growth rate function of matroid classes with exponential density that exclude specific lines, extending the understanding of their structure.

Findings

For classes excluding (q^2+1)-point lines, the growth rate is exactly (q^n-1)/(q-1) for large n.

For classes excluding (q^2+q+1)-point lines, the growth rate follows a specific shifted exponential formula.

The results classify the growth behavior of matroid classes with exponential density based on excluded lines.

Abstract

Let $\cM$ be a minor-closed class of matroids that does not contain arbitrarily long lines. The growth rate function, $h:\bN\rightarrow \bN$ of $\cM$ is given by $$h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$).$$ The Growth Rate Theorem shows that there is an integer $c$ such that either: $h(n)\le c\, n$, or ${n+1 \choose 2} \le h(n)\le c\, n^2$, or there is a prime-power $q$ such that $\frac{q^n-1}{q-1} \le h(n) \le c\, q^n$; this separates classes into those of linear density, quadratic density, and base-$q$ exponential density. For classes of base-$q$ exponential density that contain no $(q^2+1)$-point line, we prove that $h(n) =\frac{q^n-1}{q-1}$ for all sufficiently large $n$. We also prove that, for classes of base-$q$ exponential density that contain no $(q^2+q+1)$-point line, there exists $k\in\bN$ such that $h(n) = \frac{q^{n+k}-1}{q-1} - q\frac{q^{2k}-1}{q^2-1}$ for all sufficiently large $n$.