5 problems

Let $\mathcal{M}$ be a matroid and let … be two bases. Gabow's conjecture. There exist permutations $\sigma$ and $\sigma'$ such that in the sequence … every set of $r$ cyclically c…

Let $M=(S,\mathcal{B})$ be a matroid and let $S=B_1\cup\dots\cup B_k$ be a partition of the ground set into $k$ pairwise disjoint bases. Strengthened Gabow conjecture. The matroid…

Let $M=(S,\mathcal{B})$ be a matroid with rank function $r_M$. It is uniformly dense if … for every $X\subseteq S$. A matroid is cyclically orderable if its ground set has a cyclic…

Kajitani–Miyano–Ueno conjecture. The matroid $M$ is cyclically orderable if and only if

Let $M$ be a finite loopless matroid of rank $r$, and let $B=\{b_1,\dots,b_r\}$ and $B'=\{b'_1,\dots,b'_r\}$ be two bases of $M$. A cyclic ordering is an ordering considered cyclic…