2 problems

Let $(B_1,\dots,B_k)$ and $(B'_1,\dots,B'_k)$ be compatible sequences of $k$ bases of a matroid $M=(S,\mathcal{B})$, and let $w:S\to\mathbb{R}_+$. Weighted cyclic-exchange conjectu…

Let $(B_1,\dots,B_k)$ and $(B'_1,\dots,B'_k)$ be compatible sequences of $k$ bases of a rank-$r$ matroid. A cyclic exchange is the simultaneous cyclic transfer of elements among di…