36 problems

Let $\mathbb{F}$ be a finite field. An excluded minor for a class of matroids is a matroid not in the class whose proper minors all belong to the class. Rota's conjecture. The clas…

Minor-minimal obstruction conjecture. The non-half-plane-property matroids listed in the source beyond the first six are also minor-minimal, and the set of minor-minimal non-half-p…

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…

Let ${\mathfrak C}$ be a minor-closed family of binary linear codes. A code is an excluded minor of ${\mathfrak C}$ if it does not belong to ${\mathfrak C}$ while every proper mino…

Let $\Gamma$ be a finite group and let $\mathcal{M}$ be a minor-closed class of matroids, each represented by a $\Gamma$-gainable biased framework, meaning a biased framework whose…

Let $\Gamma$ be a finite group. A $\Gamma$-gainable biased graph is a biased graph admitting a gain assignment over $\Gamma$, and an excluded minor for a class is a graph outside t…

Quasi-graphic matroid conjectures. The list of excluded minors for the class of quasi-graphic matroids is finite, and there is a polynomial-time algorithm for recognition of $3$-co…

Let $\Gamma$ be a finite group, let $\Gamma_1$ be a subgroup, and let $\mathcal{A}$ be a set of groups. Finite excluded-minor conjecture. The class…

Let $\Gamma$ be a group, let $\Gamma_1$ be a subgroup, and let $\mathcal{A}$ be a set of groups such that $\Gamma/\Gamma_1$ is isomorphic to a group in $\mathcal{A}$. Infinite excl…

Rota's conjecture. For every finite field … -representability.

The excluded-minor conjecture for 2-regular matroids. This is the complete list of excluded minors for the class of 2-regular matroids; equivalently, there are no excluded minors f…

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 $H$ be a finite group, and let $\mathcal{M}$ be a minor-closed class of $H$-gain-graphic matroids. Excluded-minor conjecture. The class $\mathcal{M}$ has only finitely many exc…

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…

A gammoid is a matroid obtained from a directed graph by specifying a set of terminal vertices and taking as independent sets those linkable to the terminals by pairwise vertex-dis…

Let ${\mathbb K}$ be an infinite field, and let the class of ${\mathbb K}$-representable matroids be the minor-closed class consisting of matroids representable over ${\mathbb K}$.…

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 quasi-graphic matroid is a matroid having a framework: a graph $G$ with $E(G)=E(M)$ such that, for each component $H$ of $G$, $r(E(H))\leq |V(H)|$; for each vertex $v$,…

In [? Let $M$ be a matroid. A framework for $M$ is a graph $G$ satisfying: $E(G)=E(M)$; for each component $H$ of $G$, $r(E(H))\leq |V(H)|$; for each vertex $v\in V(G)$,…

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…