3 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 $(B_1,B_2)$ 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 $H$ be a hypergraph of $d$-intervals, let $\tau_w(H)$ be its weighted covering number, and let $\nu_w(H)$ be its weighted matching number. Weighted covering conjecture. One sho…