DeVos–Kwon–Oum branch-depth obstruction conjecture for matroids

A matroid is a matroid, and $M(F_n)$ denotes the cycle matroid of the fan $F_n$, formed from a star $K_{1,n}$ together with a path through the $n$ leaves. The uniform matroid $U_{n,2n}$ has rank $n$ on $2n$ elements. Branch-depth is the matroid parameter described in the source.

DeVos–Kwon–Oum's branch-depth obstruction conjecture. For every positive integer $n$, there is an integer $d$ such that every matroid of branch-depth at least $d$ contains a minor isomorphic to $M(F_n)$ or $U_{n,2n}$.

DeVos, Kwon, and Oum introduced branch-depth as an analogue of tree-depth for graphs. Their results show that fan matroids have branch-depth $\Theta(\log n/\log\log n)$, so they are obstructions to bounded branch-depth; the conjecture asserts that, together with the uniform matroids $U_{n,2n}$, these are the only required minor obstructions.