The girth of a matroid $M$ is the minimum size of a circuit of $M$, or $\infty$ if $M$ has no circuits. There are some uninteresting matroids of large girth: the uniform matroids $U_{t,t}$ and $U_{t-1, t}$, which are the graphic matroids of forests and cycles. However, these matroids are not cosimple; their duals contain loops or parallel pairs. Can we say anything more interesting about cosimple matroids of large girth?

Question 1: What are the unavoidable minors in cosimple matroids of large girth?

Thomassen [4] answered this question in the graphic case by showing that, surprisingly, every graph is unavoidable. A graphic matroid $M(G)$ is cosimple if and only if $G$ is $3$-edge-connected, that is, $G$ cannot be disconnected by deleting at most two edges.

Theorem 2 (Thomassen): There is a function $f$ so that for every integer $t$, every $3$-edge-connected graph of girth at least $f(t)$ contains the clique $K_t$ as a minor.

The cographic case is equivalent to a well-known theorem of Mader [3] about the average degree of graphs with a forbidden minor.

Theorem 3 (Mader): There is a function $f$ so that for every integer $t$, every cosimple cographic matroid of girth at least $f(t)$ contains $M(K_t)^*$ as a minor.

So in general we must forbid both a graphic and a cographic matroid. (There exist both graphic and cographic matroids which are cosimple and have large girth.) Geelen, Gerards, and Whittle [2] conjectured that this necessary condition is also sufficient for $GF(q)$-representable matroids. James Davies, Meike Hatzel, Kolja Knauer, Torsten Ueckerdt, and I recently proved this conjecture. The paper is out on arXiv [1].

Theorem 4 (Davies, Hatzel, Knauer, McCarty, Ueckerdt): There is a function $f$ so that for every integer $t$ and finite field $GF(q)$, every cosimple $GF(q)$-representable matroid of girth at least $f(t,q)$ contains either $M(K_t)$ or $M(K_t)^*$ as a minor.

The proof relies on interpreting the Growth Rate Theorem in terms of the shatter function of an associated set system. This allows us to use tools from the combinatorics of set systems. Many of these tools are very powerful, and yet (as far as I know), this is the first application of such tools to matroid theory. See the paper for details.

The next step would be to prove a stronger version of Theorem 4 with a line $U_{2, q+2}$ and a coline $U_{q, q+2}$ forbidden instead of the requirement about $GF(q)$-representability. The coline $U_{q, q+2}$ is a cosimple matroid of large girth. However, it is less satisfying to forbid a line since it contains short circuits.

If we do not want to forbid a line, then we must forbid a bicircular matroid. This is because there exist cosimple bicircular matroids of large girth, and for sufficiently large $t$ they do not contain $U_{t, t+2}$, $M(K_t)$, or $M(K_t)^*$ as a minor. We (rather boldly) conjecture that these are the only unavoidable cosimple matroids of large girth. Let us write $B(G)$ for the bicircular matroid of a graph $G$.

Conjecture 5 (Davies, Hatzel, Knauer, McCarty, Ueckerdt): There exists a function $f$ such that for every integer $t$, every cosimple matroid with girth at least $f(t)$ contains either $U_{q, q+2}$, $M(K_t)$, $M(K_t)^*$, or $B(K_t)$ as a minor.

References:

[1] Davies, Hatzel, Knauer, McCarty, Ueckerdt. Girth in $GF(q)$-representable matroids. ArXiv: 2504.21797, 2025.

[2] Geelen, Gerards, and Whittle. The highly connected matroids in minor-closed
classes. Ann. Comb., 19(1):107–123, 2015.

[3] Mader. Homomorphieeigenschaften und mittlere Kantendichte von Graphen.
Math. Ann., 174:265–268, 1967.

[4] Thomassen. Girth in graphs. J. Combin. Theory Ser. B, 35(2):129–141, 1983.

(Note: The theorem Thomassen proved might seem stronger at first glance since he just assumed the graph has minimum degree at least $3$. However, it is possible to reduce this theorem to the $3$-edge-connected case.)