9 problems

Let $k\geq 3$, and color the edges of the complete graph $K_{2k}$ so that each color class forms a perfect matching. A spanning tree is rainbow colored if no two of its edges have…

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

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 $B$.

Let $K$ be an absolute constant, and let $E_{I,j}$ denote the edge set of color class $j$, for $j=1,2,\ldots,s$, in the random graph process on $K_n$. A MultiHamiltonCycle is a set…

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

Let ${\mathcal M}$ and ${\mathcal N}$ be two matroids on the same vertex set. For a family $F_1,\ldots,F_k$, an $({\mathcal M}\cap{\mathcal N})$-SR is a choice of one element from…