4 problems

Let $\mathcal{M}=(E,\mathcal{I})$ be a matroid whose ground set $E$ can be partitioned into $k$ disjoint bases, and let $S_1,S_2,\ldots,S_\ell$ be pairwise disjoint subsets of $E$.…

Let $M=(S,\mathcal{B})$ be a matroid and let $S=B_1\cup\dots\cup B_k$ be a partition of the ground set into $k$ pairwise disjoint bases. Strengthened Gabow conjecture. The matroid…

Let $M$ be a matroid whose ground set $E$ can be partitioned into $k$ disjoint bases. The matroid is equitable if, for every subset $Z\subseteq E$, there is a partition…

Equitability conjecture. If the ground set $S$ of $M$ can be partitioned into two bases, then $M$ is equitable. This is a simple but surprisingly open question about equitable matr…