$\newcommand{\cG}{\mathcal{G}}$ $\newcommand{\cM}{\mathcal{M}}$ $\newcommand{\elem}{\varepsilon}$ $\newcommand{\con}{/}$ $\newcommand{\del}{\!\backslash\!}$

I’m going to give what is hopefully a less frenetically paced exposition of what many readers have probably seen me talk about more that once – the growth rates of matroids in minor-closed classes. I’ll try to cover things from the ground up in order to motivate the subject, as well as provide some short proofs. The traditional starting point, from which I won’t deviate, is a theorem of Mader [Mader67]:

Theorem 1. If $\mathcal{G}$ is a proper minor-closed class of graphs, then $|E(G)| \le c_{\mathcal{G}}|V(G)|$ for each simple graph $G$ in $\cG$.

Here, `proper’ means that $\cG$ isn’t the class of all graphs, and $c_{\mathcal{G}}$ is a positive real constant depending only on $\cG$. Note that this theorem fails if we allow $\mathcal{G}$ to be the class of all graphs; the complete graph $K_t$ has $\binom{t}{2}$ edges and $t$ vertices, and the ratio $\binom{t}{2}$ is not bounded above by any linear function of $t$. With this fact in mind, what is interesting is the dichotomy; either $\cG$ is the class of all graphs and the densest simple graphs in $\mathcal{G}$ (the cliques) have a number of edges that is quadratic in their number of vertices, or $\mathcal{G}$ is not the class of all graphs and this bound is linear. This is not just a curiosity – we know now that it reflects the structure proved by Robertson and Seymour in the graph minors structure theorem – the graphs in a proper minor-closed class are at most linearly dense because they are `almost’ embeddable on a fixed surface. Of course we didn’t know this in 1967, so it presented a question rather than a corroboration.

Moving on to matroids, we first define some convenient notation. For a matroid $M$, we write $\elem(M)$ for $|\mathrm{si}(M)|$ (equivalently the number of rank-$1$ flats of $M$); this stops us having to awkwardly talk about simple matroids all the time. For a minor-closed class $\cM$, we use $h_{\cM}(n)$ to denote the function whose value at an integer $n$ is the maximum of $\elem(M)$ over all matroids $M \in \cM$ whose rank is at most $n$. Some people call this the size function but I’ll call in the growth rate function. We allow $h_{\cM}(n) = \infty$ – indeed any class that contains $U_{2,k}$ for all $k$ will satisfy this for all $n \ge 2$. In the interesting cases this function is monotonely increasing without bound. For example, for the class $\cM$ of graphic matroids has growth rate function $h_{\cM}(n) = \binom{n+1}{2}$, and by Theorem 1 above, all of its proper subclasses have growth rate function at most linear in $n$.

The next few theorems will combine to divide minor-closed classes of matroids into four types, just as Theorem 1 divides classes of graphs into two. These theorems all concern `dichotomies’ in growth rate functions. The first, due to Kung [Kung91] distinguishes classes with finite growth rate functions from those with infinite growth rate function. We’ll even prove it, since the proof is so nice.

Theorem 2.  If $\cM$ is a minor-closed class of matroids, then either

• $h_{\cM}(n) = \infty$ for all $n \ge 2$, or
• there is some $\ell \ge 2$ such that $U_{2,\ell+2} \notin \cM$ and $h_{\cM}(n) \le \frac{\ell^n-1}{\ell-1}$ for all $n$.

Proof: If $U_{2,\ell+2} \in \cM$ for all $\ell \ge 2$ then the first outcome holds, so we may assume that $U_{2,\ell+2} \notin \cM$ for some $\ell \ge 2$, and will show that $\elem(M) \le \frac{\ell^r-1}{\ell-1}$ for each integer $r$. Let $M \in \cM$ be a rank-$r$ matroid and $e$ be a nonloop of $M$. Inductively we may assume that $\elem(M \con e) \le \frac{\ell^{r-1}-1}{\ell-1}$, and we know $M$ has $\elem(M \con e)$ lines containing $e$. Since $M$ has no $U_{2,\ell+2}$-minor, each of these lines has at most $\ell$ points other than $e$, so $\elem(M) \le \ell \elem(M \con e) + 1 \le \ell\frac{\ell^{r-1}-1}{\ell-1}+ 1 \le \frac{\ell^r-1}{\ell-1}$, as required.

This theorem tells us that there is no minor-closed class with finite growth rate function greater than exponential in $n$, and cleanly divides growth rate functions into the infinite and the finite. In fact, there is much more to the story than this; we’ll finish by stating the theorem that will be the focus of my next post. This theorem was conjectured by Kung [Kung91] and stated in [GKW09] as a consequence of various theorems of Geelen, Kabell, Kung and Whittle.

Growth Rate Theorem. If $\cM$ is a minor-closed class of matroids, then either

• $h_{\cM}(n) \le c_{\cM}n$ for all $n$,
• $\binom{n+1}{2} \le h_{\cM}(n) \le c_{\cM}n^2$ for all $n$ and $\cM$ contains all graphic matroids,
• there is a prime power $q$ such that $\frac{q^n-1}{q-1} \le h_{\cM}(n) \le c_{\cM}q^n$ for all $n$ and $\cM$ contains all GF$(q)$-representable matroids, or
• $h_{\cM}(n) = \infty$ for all $n \ge 2$ and $\cM$ contains all simple rank-$2$ matroids.

Again $c_{\cM}$ is always a real constant depending only on $\cM$. In other words, up to a constant, there are just four possible qualitative behaviours for growth rate functions: linear, quadratic, exponential and infinite. This is dramatic behaviour and deserves more consideration. See you next time!

References

• [GKW09] J. Geelen, J. P. S. Kung, and G. Whittle. Growth rates of minor-closed classes of matroids. J. Combin. Theory Ser. B, 99(2):420–427, 2009.
• [Kung91] J. P. S. Kung. Extremal matroid theory. In Graph structure theory (Seattle, WA, 1991), volume 147 of Contemp. Math., pages 21–61. Amer. Math. Soc., Providence, RI, 1993.
• [Mader67] W. Mader. Homomorphieeigenschaften und mittlere kantendichte von graphen. Mathematische Annalen, 174:265–268, 1967. 10.1007/BF01364272.