7 problems

Rota's conjecture. For each finite field $F$, there are, up to isomorphism, only finitely many excluded minors for the class of $F$-representable matroids.

Let $ll,t in N$ with $ll ge 3$, and let $\falM$ be the class of $R$-representable matroids with no $U_{2,ll+2}$-minor and no $M(K_t)$-minor. Here $\alpha$ is the constant appearing…

Let $M$ be a representable matroid with set of circuits $\mathcal{C}$. For matroids on a fixed ground set, use the weak order defined by $M_1\geq M_2$ when every dependent set of…

The asymptotic enumeration conjecture. Asymptotically almost all representable matroids on $[n]$ have rank in $\{\lfloor n/2\rfloor,\lceil n/2\rceil\}$ and exactly $d(n)-1$ nonbase…

A representable matroid is a matroid satisfying the rank axioms of a matroid and representable over a field. Mayhew–Newman–Whittle conjecture. It is impossible to characterise the…

For a matroid $M$ on ground set $E(M)$, let $Q,R,S,T\subseteq E(M)$ satisfy $Q\cap R=S\cap T=\emptyset$. Write … Let $c:\mathbb{N}^2\to\mathbb{N}$ be a function. Geelen's conjectur…

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…