6 problems

Johnson–Robertson–Seymour's branch-width obstruction conjecture. For every positive integer $n$, there is an integer $d$ such that every matroid of branch-width at least $d$ contai…

Let $M_1$ be a matroid with a circuit-hyperplane $H$, and let $M_2$ be obtained from $M_1$ by relaxing $H$. If both $M_1$ and $M_2$ are representable over a finite field…

Matroid structure theorem. The stated tree-decomposition exists.

Let $q$ be a prime power, let $N$ be a matroid representable over $\mathrm{GF}(q)$, and call a matroid $\\{N\\}$-fragile if, for every element $e$, at least one of its deletion or…

Let $N$ be a $\operatorname{GF}(q)$-representable matroid, and let $\operatorname{bw}(M)$ denote the branch width of a matroid $M$. High-branch-width non-fragility conjecture. Ther…

Fix a matroid $N$. A matroid $M$ is $N$-fragile if, for every $e\in E(M)$, at least one of $M{-} e$ and $M/e$ has no $N$-minor. Bounded Canopy Conjecture. For every prime power $q$…