2 problems

Let $(B_1,B_2)$ and $(B'_1,B'_2)$ be compatible basis pairs of a matroid $M=(S,\mathcal{B})$, and let $w:S\to\mathbb{R}_+$. The weight of a symmetric exchange replacing $B_1,B_2$ b…

Let $(B_1,\dots,B_k)$ and $(B'_1,\dots,B'_k)$ be sequences of $k$ bases of a matroid. Two sequences are compatible if the union of the $B_i$ as a multiset equals the union of the…