20 problems

Let $\mathrm{M}$ be a matroid, and let $\{B_1,B_2\}$ and $\{B_1',B_2'\}$ be unordered pairs of bases such that the multiset union of $B_1$ and $B_2$ is equal to the multiset union…

Let $M$ be a matroid of rank $r$ with two disjoint bases $A$ and $B$. A symmetric exchange chooses $e\in A\setminus B$ and $f\in B\setminus A$ such that both $A-e+f$ and $B-f+e$ ar…

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 $M$ be a matroid of rank $r$, and let $(A,B)$ and $(A',B')$ be compatible basis pairs, meaning that … A symmetric exchange is an exchange of elements between the two bases that…

A rank-$r$ matroid is subsequence-interchangeably base orderable (SIBO) if every pair of bases $A$ and $B$ admits orderings $a_1,\ldots,a_r$ of $A$ and $b_1,\ldots,b_r$ of $B$ such…

Let $M$ be a matroid with ground set $E$. A basis sequence is a finite sequence of bases of $M$. Two basis sequences are equivalent if one can be obtained from the other by a compo…

Let $\omega\colon \binom{E}{r}\to\mathbb{R}\cup\{-\infty\}$ be a valuated matroid of rank $r$ on a ground set $E$. The source's preceding discussion defines the desired exchange pr…

Let $A$ and $B$ be bases of a matroid $\mathbf{M}$ with rank function $r$. For any $X\subseteq A\setminus B$ and $Y\subseteq B\setminus A$ and integer $p$ with $|Y| \le p \le |X|$,…

Let $\mathcal{M}$ be a matroid, and let ${\cal X}=(X_1,\ldots,X_k)$ and ${\cal Y}=(Y_1,\ldots,Y_k)$ be two collections of $k$ bases. A symmetric exchange replaces two bases…

A matroid is subsequence-interchangeably base orderable (SIBO) if every pair of bases admits the subsequence exchange orderings specified in the paper. Sparse-paving SIBO conjectur…

Let $M$ be a matroid with ground set $E$. For $k\in\mathbb{N}$, let $\psi_i\colon E\to\Gamma_i$ be group labelings into abelian groups $\Gamma_i$, let $f_i\in\Gamma_i$ be forbidden…

Let $M$ be a matroid, $psi\colon E \to \Gamma$ a labeling from the ground set $E$ of $M$ to an abelian group $\Gamma$, and $F \subseteq \Gamma$ a finite collection of forbidden lab…

Let $M$ be a matroid with ground set $E$, let $\Gamma$ be an abelian group, let $\psi\colon E\to\Gamma$ be a group labeling, and let $F\subseteq\Gamma$ be a finite set of forbidden…

Let $X_1$ and $X_2$ be bases of a matroid $M$. Subsets $A_1\subseteq X_1$ and $A_2\subset X_2$ are serially exchangeable if there are orderings $A_1=\\{a^1_1,\dots,a^1_q\\}$ and…

Weighted exchange-distance conjecture. The weighted exchange distance of $\mathbf{P}_1$ and $\mathbf{P}_2$ is at most

White's conjecture. Two basis pairs $\mathbf{P}_1$ and $\mathbf{P}_2$ are equivalent if and only if they are compatible.

Common-basis conjecture. There exist partitions of $E\setminus\{x\}$ and $E\setminus\{y\}$ into $k$ pairwise disjoint bases that share a common basis.

White's intermediate conjecture. The relations $\sim_1$ and $\sim_3$ are equal.

Cyclic ordering conjecture. It is possible to place the elements of the ground set of $M$ on a circle so that every $r(M)$ consecutive elements form a basis of $M$.

Farber's conjecture. The basis pair graph of any matroid is connected.