9 problems

Branch-depth minor-closure conjecture. There exists a function $f$ such that, if a matroid $M$ has branch-depth $d$, then there exists a matroid $N$ with contraction-deletion-depth…

A matroid is sparse paving if its nonspanning circuits are hyperplanes; equivalently, no nonbasis of size equal to the rank can be transformed into another nonbasis by exchanging o…

Forbidden-minor conjecture. A regular oriented matroid $M$ is non-even if and only if none of its $\mathsf{GB}$-minors is isomorphic to $M^*(\vec{K}_{m,n})$ for some $m,n\geq 2$ su…

Let $M$ be a connected $k$-polymatroid and let $N$ be a connected minor of $M$. A connected minor-preserving deletion or contraction is a deletion or contraction that remains conne…

Six-element-minor conjecture. The oriented matroid ${\mathcal{M}}$ is pure if and only if all of its six-element minors are pure. Equivalently, ${\mathcal{M}}$ is pure if and only…

Connectivity dichotomy conjecture. Let $k,l\in\mathbb{N}$. Asymptotically almost all matroids without a $U_{k,2k}$-minor are either vertically $l$-connected or cyclically $l$-conne…

Sparse paving minor conjecture for sparse paving matroids. Let $N$ be a fixed sparse paving matroid. Asymptotically almost every sparse paving matroid has an $N$-minor.

Sparse paving minor conjecture. Let $N$ be a fixed sparse paving matroid. Asymptotically almost every matroid has an $N$-minor.

A triangle in a matroid is a circuit of size $3$. A matroid admits another matroid as a minor if that minor is obtained by deletions and contractions. Three-triangle matroid minor…