$\newcommand{\PG}{\text{PG}}$

**Denser than a Geometry**

Hello everyone. In my last post, I discussed an unavoidable minor theorem for large matroids of density greater than $\binom{n+1}{2}$, which as a consequence characterised exactly the minor-closed classes of matroids that grow like the the graphic matroids. Today I’ll write about the exponential analogue of this problem; this appears in a paper of mine [2] available on the arXiv, and I will also be speaking on this topic at the matroid session of the upcoming CanaDAM conference in Saskatchewan.

I’ll first state a nice theorem of Kung, specialized to prime powers:

**Theorem: **If $q$ is a prime power, and $M$ is a simple rank-$n$ matroid with $|M| > \frac{q^n-1}{q-1}$, then $M$ has a $U_{2,q+2}$-minor.

This is certainly an important theorem – it says that if $M$ has too many elements to be a projective geometry over GF$(q)$, then $M$ has a rank-$2$ minor with the same property. However, if $n$ is large then this is somewhat unsatisfying, as the $U_{2,q+2}$-minor has constant size, and does not really ‘explain’ why $M$ is so dense. We would like a similar theorem that, instead of finding a $U_{2,q+2}$-minor, finds an unavoidable minor whose size grows with $n$.

What should such a minor be? That is, what are the large matroids that are ‘canonically’ denser than a projective geometry over GF$(q)$?

The first example is easy; rank-$2$ matroids can contain arbitrarily many elements, so the uniform matroid $U_{2,t}$ for some large $t$ must appear as an outcome. The direct sum of $U_{2,q^n}$ and $U_{n-2,n-2}$, for example, is a rank-$n$ matroid with more elements than $\PG(n-1,q)$, and essentially no more interesting structure.

Another matroid denser than $\PG(n-1,q)$ is simply a projective geometry $\PG(t-1,q’)$ over finite field GF$(q’)$.with $q’ > q$. This is our second outcome.

The next two outcomes are more interesting. An obvious way to construct a matroid denser than $\PG(n-1,q)$ is to perform a single-element extension. By modularity, every extension of $\PG(n-1,q)$ consists of adding a point freely to some flat $F$. The smallest simple case is where $F$ is a line: we write $\PG^{\mathrm{line}}(n-1,q)$ for the matroid formed by adding a point freely to a line of $\PG(n-1,q)$. The ‘largest’ case is when $F$ is the entire ground set, so the extension is free. We write $\PG^{\mathrm{free}}(n-1,q)$ for this matroid. The matroids $\PG^{\mathrm{line}}(m-1,q)$ and $\PG^{\mathrm{free}}(n-1,q)$ coincide for $n = m = 2$, but are not minors of eachother for any $m,n \ge 3$; they will thus occur as separate outcomes in our theorem. On the other hand, it is not hard to show that any simple extension of $\PG(2n-1,q)$ has one of $\PG^{\mathrm{line}}(n-1,q)$ or $\PG^{\mathrm{free}}(n-1,q)$ as a minor; thus, no further single-element extensions are needed as outcomes. Here is the theorem.

**Theorem 1 [2]: **For every prime power $q$ and all $t \ge 3$, there exists $n \in \mathbb{Z}$ so that, if $M$ is a simple matroid with rank at least $n$ and $|M| > \frac{q^{r(M)}-1}{q-1}$, then $M$ has a minor isomorphic to one of the following:

- $U_{2,t}$,
- $\PG(t-1,q’)$ for some $q’ > q$,
- $\PG^{\mathrm{free}}(t-1,q)$, or
- $\PG^{\mathrm{line}}(t-1,q)$.

Like in the quadratic case, we can apply this theorem to identify minor-closed classes that grow no faster than the GF$(q)$-representable matroids. For instance, if $q’$ is the next prime power after $q$, and $\ell$ is an integer so that $q \le \ell < q’$, then, for $t = \ell+2$, all four of the above minors can be shown to contain a $U_{2,\ell}$-minor. We thus have the following:

**Corollary 1: **Let $q$ and $q’$ be consecutive prime powers and $\ell$ be an integer so that $q \le \ell < q’$. If $M$ is a simple matroid with sufficiently large rank and no $U_{2, \ell+2}$-minor, then $|M| \le \frac{q^{r(M)}-1}{q-1}$.

**Spikes and Swirls**

Let $X = \{x_1,\dotsc,x_t\}$ and $Y = \{y_1, \dotsc, y_t\}$ be disjoint $t$-element sets. A rank-$t$ *free spike *is the matroid $\Lambda_t$ with ground set $X \cup Y$ whose non-spanning circuits are exactly the four-element sets of the form $\{x_i,x_j,y_i,y_j\}$ for $i \ne j$. The rank-$t$ *free swirl *is the matroid $\Delta_t$ with ground set $X \cup Y$ whose non-spanning circuits are exactly the four-element sets $\{x_i,x_{i+1},y_i,y_{i+1}\}$, where addition is modulo $t$. Both free spikes and free swirls arise as pathologies in matroid representation theory.

Let $q \ge 3$ be a prime power and let $t \ge 3$. One can show (see [1]) that

- $\Lambda_t$ is GF$(q)$-representable if and only if $q$ is non-prime or $t \le q-2$.
- $\Delta_t$ is GF$(q)$-representable if and only if $q-1$ is non-prime or $t \le q-3$.

Using this, one can hunt for free spikes and swirls in the list of unavoidable minors in Theorem 1. There is a bit of case analysis to see what theorems fall out, but we get a few different corollaries that give upper bounds on the densities of matroids with out line-, spike-, and/or swirl-minors. All of the upper bounds in these corollaries are the densities of projective geometries, and are best-possible because the geometries themselves are tight examples.

**Corollary 2: **Let $\ell \ge 2$ and $t \ge 3$ be integers. If $M$ is a matroid of sufficiently large rank with no $U_{2,\ell+2}$-minor and no $\Lambda_t$-minor, then $|M| \le \frac{p^{r(M)}-1}{p-1}$.

**Corollary 3: **Let $\ell \ge 3$ and $t \ge 3$ be integers. If $M$ is a matroid of sufficiently large rank with no $U_{2,\ell+2}$-minor, no $\Lambda_t$-minor and no $\Delta_t$-minor, then $|M| \le \tfrac{1}{2}(3^{r(M)}-1)$.

The next corollary mostly tells you how dense a matroid can be if you exclude a line and a free swirl has a minor. However, it weirdly fails to apply to some $\ell$ because of the oddness of representation of free swirls. The only fields that can’t represent free swirls are are the ones of order $2^p$, where $2^p-1$ is prime. A prime of the form $2^p-1$ for another prime $p$, like $2^3 – 1 = 7, 2^5 – 1 = 31$ or $2^7 – 1 = 127$, is a *Mersenne prime; *it is not even known if there are infinitely many.

**Corollary 4: **Let $2^p-1$ and $2^{p’}-1$ be consecutive Mersenne primes, and let $k$ and $\ell$ be integers so that $2^p \le \ell < \min(2^{2p} + 2^p, 2^{p’})$ and $k \ge \max(3,2^p-2)$. If $M$ is a simple matroid of sufficiently large rank with no $U_{2,\ell+2}$ or $\Delta_k$-minor, then $|M| \le \frac{2^{pr(M)}-1}{2^p-1}$.

This fails to apply to some $\ell$, but only if $\ell$ is (roughly) greater than the square of a Mersenne prime but smaller than the next Mersenne primes. The first pair of Mersenne primes this far apart are $2^{127}-1$ and $2^{521}-1$, so for some $\ell$ around this size, the above corollary says nothing. The actual fact of the matter depends on the very poorly understood distribution of the Mersenne primes, so I don’t expect much improvement there in the near future.

Thanks, and see you next time!

[1] J Geelen, J.G. Oxley, D. Vertigan, G. Whittle, Totally free expansions of matroids, J. Combin. Theory. Ser. B 84 (2002), 130-179.

[2] P. Nelson, Matroids denser than a projective geometry, to appear, SIAM Journal on Discrete Mathematics.