Odd circuits in dense binary matroids

We show that, for each real number $\alpha > 0$ and odd integer $k\ge 5$ there is an integer $c$ such that, if $M$ is a simple binary matroid with $|M| \ge \alpha 2^{r(M)}$ and with no $k$-element circuit, then $M$ has critical number at most $c$. The result is an easy application of a regularity lemma for finite abelian groups due to Green

Projective geometries in exponentially dense matroids. I

We show for each positive integer $a$ that, if \cM is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $n$ such that either every rank-$r$ matroid in \cM can be covered by at most $r^n$ sets of rank at most $a$, or \cM contains the \GF(q)-representable matroids for some prime power $q$, and every rank-$r$ matroid in \cM can be covered by at most $r^nq^r$ sets of rank at most $a$. This determines the maximum density of the matroids in \cM up to a polynomial factor

On minor-closed classes of matroids with exponential growth rate

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 $n$