17 problems

Let ${\mathfrak C}$ be a minor-closed class of codes over a finite field ${\mathbb F}$. For a collection ${\mathcal F}$ of codes, let ${\mathfrak C}_{\mathcal F}$ denote the class…

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 $M$ be a matroid. A matroid is $3$-regular if it has a representation over $\mathbb{Q}(\alpha_1,\alpha_2,\alpha_3)$ in which every non-zero subdeterminant is an integer power o…

Finite excluded-minor conjecture. There are, up to isomorphism, only finitely many excluded minors for the class of quasi-graphic matroids.

Let $H$ be a finite group, and let $\mathcal{M}$ be any minor-closed class of $H$-gain-graphic matroids. A gain-graphic matroid is a frame matroid obtained from a graph whose edges…

Let $\mathcal M$ be a class of matroids. An antichain is a collection of matroids in which no member is a minor of another. The class $\mathcal M$ is weakly fractal when the sequen…

Let ${\mathcal M}$ be a minor-closed class of matroids, and let $\Gamma_{\mathcal M}(n)$ denote the proportion of $n$-element matroids in the class's boundary, namely its excluded…

For a matroid $M$, let $\d(M)$ denote its density, and call $M$ strictly $t$-density-critical when $d(M)>t$ while every proper minor $N$ of $M$ satisfies $d(N)\leq t$. Finiteness c…

Let $M$ be a matroid. A collection of independent sets covers $M$ if every element of $M$ belongs to at least one set in the collection. Finiteness conjecture. For every positive i…

A matroid $M$ is a frame matroid if there is a matroid $M'$ with a basis $V$ such that … and, for each $e\in E(M)$, the unique circuit in $V\cup\{e\}$ has size at most $3$. An excl…

Let $BL$ be the class of bicircular lift matroids, and let $\overline{BL}$ be the union of $BL$ with the class of graphic matroids. The class $\overline{BL}$ is minor-closed. Pivot…

A matroid $M$ is 1-flowing if it is $e$-flowing for every element $e\in E(M)$, where $M$ is $e$-flowing when, for every assignment of non-negative integral capacities to the elemen…

Obstruction excluded-minor conjecture. If $\Delta$ has an obstruction, then $M(\Delta)$ is an excluded minor for $\mathcal{BO}$.

Ingleton's critical-graph conjecture. If $\Delta$ is a critical graph with no obstructions, then $M(\Delta)$ is an excluded minor for $\mathcal{BO}$.

Minor-minimality conjecture. For all integers $n\geq 0$, $M\!I_n$ are minor-minimal with respect to being non-weakly-orientable. Hence, weak-orientability cannot be described by a…

Let $\mathbb{P}$ be a partial field, and let $\operatorname{\mathcal{M}}(\mathbb{P})$ denote the class of $\mathbb{P}$-representable matroids. Partial-field inequivalent-representa…

A partial field $\mathbb{P}$ is finitary if it admits a homomorphism to $\operatorname{GF}(q)$ for some prime power $q$. Let $\operatorname{\mathcal{M}}(\mathbb{P})$ denote the cla…