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

Projective geometries in exponentially dense matroids. II

We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $r^c$ rank-$a$ sets, or $\mathcal{M}$ contains the GF$(q)$-representable matroids for some prime power $q$ and every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $cq^r$ rank-$a$ sets. In the latter case, this determines the maximum density of matroids in $\mathcal{M}$ up to a constant 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 $n$

The densest matroids in minor-closed classes with exponential growth rate

The $\mathit{growth\ rate\ function}$ for a nonempty minor-closed class of matroids $\mathcal{M}$ is the function $h_{\mathcal{M}}(n)$ whose value at an integer $n \ge 0$ is defined to be the maximum number of elements in a simple matroid in $\mathcal{M}$ of rank at most $n$. Geelen, Kabell, Kung and Whittle showed that, whenever $h_{\mathcal{M}}(2)$ is finite, the function $h_{\mathcal{M}}$ grows linearly, quadratically or exponentially in $n$ (with base equal to a prime power $q$), up to a constant factor. We prove that in the exponential case, there are nonnegative integers $k$ and $d \le \tfrac{q^{2k}-1}{q-1}$ such that $h_{\mathcal{M}}(n) = \frac{q^{n+k}-1}{q-1} - qd$ for all sufficiently large $n$, and we characterise which matroids attain the growth rate function for large $n$. We also show that if $\mathcal{M}$ is specified in a certain `natural' way (by intersections of classes of matroids representable over different finite fields and/or by excluding a finite set of minors), then the constants $k$ and $d$, as well as the point that `sufficiently large' begins to apply to $n$, can be determined by a finite computation

Growth rate functions of dense classes of representable matroids

AbstractFor each proper minor-closed subclass M of the GF(q2)-representable matroids containing all 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 M, and give a rank-r matroid in M for which equality holds. As a consequence, we give a tight upper bound on the number of points in a GF(q2)-representable, rank-r matroid of large rank with no PG(k,q2)-minor

Exponentially Dense Matroids

This thesis deals with questions relating to the maximum density of rank-n matroids in a minor-closed class. Consider a minor-closed class M of matroids that does not contain a given rank-2 uniform matroid. The growth rate function is defined by h_M(n) = max(|N| : N ∈ M simple, r(N) ≤ n). The Growth Rate Theorem, due to Geelen, Kabell, Kung, and Whittle, shows that the growth rate function is either linear, quadratic, or exponential in n. In the case of exponentially dense classes, we conjecture that, for sufficiently large n, h_M(n) = (q^(n+k) − 1)/(q-1) − c, where q is a prime power, and k and c are non-negative integers depending only on M. We show that this holds for several interesting classes, including the class of all matroids with no U_{2,t}-minor. We also consider more general minor-closed classes that exclude an arbitrary uniform matroid. Here the growth rate, as defined above, can be infinite. We define a more suitable notion of density, and prove a growth rate theorem for this more general notion, dividing minor-closed classes into those that are at most polynomially dense, and those that are exponentially dense

Quadratically Dense Matroids

This thesis is concerned with finding the maximum density of rank-$n$ matroids in a minor-closed class. The extremal function of a non-empty minor-closed class $\mathcal M$ of matroids which excludes a rank-2 uniform matroid is defined by $$h_{\mathcal M}(n)=\max(|M|\colon M\in \mathcal M \text{ is simple, and } r(M)\le n).$$ The Growth Rate Theorem of Geelen, Kabell, Kung, and Whittle shows that this function is either linear, quadratic, or exponential in $n$. In this thesis we prove a general result about classes with quadratic extremal function, and then use it to determine the extremal function for several interesting classes of representable matroids, for sufficiently large integers $n$. In particular, for each integer $t\ge 4$ we find the extremal function for all but finitely many $n$ for the class of $\mathbb C$-representable matroids with no $U_{2,t}$-minor, and we find the extremal function for the class of matroids representable over finite fields $\mathbb F_1$ and $\mathbb F_2$ where $|\mathbb F_1|-1$ divides $|\mathbb F_2|-1$ and $|\mathbb F_1|$ and $|\mathbb F_2|$ are relatively prime