11 problems

Short rainbow circuit conjecture. If $(M,c)$ has no colour-singular elements, then $M$ has a short rainbow circuit. This conjecture extends the corresponding rank-two case from gra…

Let $M$ be a simple regular matroid, let $r(M)$ denote its rank, and let $c$ colour the elements of $M$ with $r(M)+1$ colours, with each colour class containing at least two elemen…

Let $M$ be an oriented regular matroid, equipped with triangulating circuit and cocircuit signatures. A consistent sandpile torsor algorithm is a consistent algorithm assigning san…

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…

Let $M=(E,\mathcal B,\chi)$ be an oriented regular matroid, let $(\sigma,\sigma^*)$ be a pair of acyclic signatures, and let $\alpha$ and $\beta$ be matroidal sandpile torsor algor…

Let $M=(E,\mathcal B,\chi)$ be an oriented regular matroid, let $(\sigma,\sigma^*)$ be a pair of acyclic circuit and cocircuit signatures, and let $\alpha_{(M,\sigma,\sigma^*)}$ be…

Gedeon's non-degeneracy conjecture. Every connected regular matroid is non-degenerate. The conjecture remains open; the source emphasizes that regular matroids form a highly restri…

Regular-matroid Jacobian exponent conjecture. For every positive integer $k$, there are only finitely many connected regular matroids $M$ such that the exponent of…

Let $M$ be a simple matroid of rank $n-1$, and let $c$ be a colouring of its ground set $E(M)$ with $n$ colours. A circuit is rainbow when no two of its elements have the same colo…

Let $\mathcal{M}$ be a regular matroid and let $M$ be a unimodular $m \times n$ representation of $\mathcal{M}$ with corank greater than $1$. Let $L$ denote the associated matrix w…

Bounded-rank conjecture. Then