5 problems

Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be matroids on the same ground set $E$, and assume that $E$ can be partitioned into $k$ independent sets in each matroid. For a set…

Equivalent Rainbow Arborescence Conjecture. If $k \ge n-1$, then the disjoint union $G$ of $k$ spanning arborescences has a rainbow spanning arborescence $B$.

Yokoi's Rainbow Arborescence Conjecture. If $G$ is the disjoint union of $n-1$ spanning arborescences $A_1, A_2, \dots, A_{n-1}$, then $G$ has a rainbow spanning arborescence $B$.

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

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…