Matroids denser than a clique

TL;DR

This paper characterizes minor-closed matroid classes whose maximum size matches the number of edges in a complete graph for large ranks, extending understanding of growth-rate functions in matroid theory.

Contribution

It provides a complete characterization of classes with quadratic growth-rate functions equal to the binomial coefficient, refining the growth-rate theorem for matroids.

Findings

Classes with quadratic growth-rate functions equal to binomial coefficients are characterized.

Such classes necessarily contain all graphic matroids.

The paper extends the understanding of growth-rate functions in minor-closed matroid classes.

Abstract

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential, or infinite. Morover, if the growth-rate function is quadratic, then $h(r)\ge \binom{r+1}{2}$, with the lower bound coming from the fact that such classes necessarily contain all graphic matroids. We characterise the classes that satisfy $h(r) = \binom{r+1}{2}$ for all sufficiently large $r$.