15 problems

Let $E$ be a ground set, let $\mathfrak{F}(E)$ denote the relevant class of matroids on $E$, and for $j\in\{0,1\}$ let $\mathcal{I}_{M_j}$ be the collection of independent sets of…

Nash-Williams' Matroid Intersection Conjecture. The matroids $M$ and $N$ admit a common independent set $I$ for which there is a partition

Existence conjecture. The class of wild orthogonally oriented matroids exists.

Let $M$ and $N$ be finitary matroids on a common ground set. For a set $I$, write $M/I$ and $N/I$ for the matroid contractions, and let $r(M/I)$ and $r(N/I)$ denote their ranks. An…

Multi-Matroid Intersection Conjecture. The matroids admit common independent sets $I_1,\dots,I_{k-1}$ such that, for each $1\leq i\leq k-1$, there is a partition

Packing/Covering Conjecture. For every family $(M_i:i\in\Theta)$ of matroids on a common edge set $E$, there is a partition

Nearly thin duality conjecture. Matroids from $n$-nearly thin families are dual to $n$-nearly finitary matroids.

The paper studies matroids over skew hyperfields and the equivalence between their circuit axioms, quasi-Plücker axioms, and dual signatures. Extension conjecture. There is no real…

Complementarity conditions conjecture. There always exist an $(I,t)$-linkage $\mathcal{P}$ and a vertex set $X\ni t$ such that $\mathcal{P}$ and $X$ satisfy the complementarity con…

Let $M$ and $N$ be tame matroids on the same ground set $E$. A packing for a pair of matroids on a set is a pair of disjoint spanning sets, one for each matroid, and a covering is…

Let $(D,B_0)$ be a dimaze, and let $M_L(D,B_0)$ be its linkability system. For every $I\in M_L(D,B_0)$ and every $B\subseteq B_0$, suppose there is a maximal independent set in…

Let $\mathcal{T}$ be a nice tree, and let $M_{\emptyset}(\mathcal{T})$ and $M_{\Omega}(\mathcal{T})$ be the matroids obtained by gluing finite matroids along $\mathcal{T}$, allowin…

Let $(M_k\mid k\in K)$ be a family of matroids on the same ground set $E$. A covering for $(M_k.Y\mid k\in K)$ is a family of independent sets whose union covers $Y$, and a packing…

Let $(M_k\mid k\in K)$ be a family of matroids. A wave is a set together with disjoint spanning sets for the restrictions of the matroids to that set; a family is unhindered when i…

Let $(M_k\mid k\in K)$ be a family of matroids on the same ground set $E$. A packing consists of pairwise disjoint spanning sets, one for each $M_k$, and a covering consists of ind…