12 problems

Direct-sum conjecture for transversal q-matroids. The direct sum $M_1\oplus M_2$ is a transversal $q$-matroid with presentation $\mathcal{X}_1\oplus\mathcal{X}_2$.

Let $[n]=\{1,2,\ldots,n\}$, and let $M[(S_1,S_2,\dots,S_k)]$ be the transversal matroid presented by subsets $S_1,S_2,\dots,S_k\subseteq[n]$. A set system $(S_1,S_2,\dots,S_k)$ is…

Acketa's conjecture. The lattices $L_5$, $L_6$, and $L_7$ are Tr-lattices.

Rank-three transversal matroid conjecture. Every rank-$3$ transversal matroid has the half-plane property.

Strongly base-orderable matroid conjecture. Every strongly base-orderable matroid has the half-plane property.

Let $M$ be a $q$-transversal matroid with a presentation by a system of subspaces. A presentation is minimal if no member can be replaced by a proper subspace while preserving the…

Let $M$ be a $q$-matroid on a vector space $V$, with bar nullity function $\overline{n}$, and let $\mathcal X$ be a system of subspaces of $V$. A q-transversal is a subspace obtain…

Rank-three transversal matroid conjecture. Every transversal matroid of rank $3$ has the half-plane property.

Let $\mathcal{M}$ be the class of matroids that are both transversal and cotransversal. A matroid class is strongly pigeonhole if, for every $\lambda$, there is a bound depending o…

Let $\mathcal{B}$ be a transversal positroid. For $a\in[n]$, write $<_a$ for the cyclic order beginning at $a$, and order equal-sized sets by the corresponding $a$-Gale order: if…

Let $\mathcal{B}(\mathcal{S})$ be the transversal matroid presented by a set system $\mathcal{S}$. A pivot is the presentation operation defined in the source that replaces one mem…

Stiefel-map determination conjecture. A linkage matching field is determined by the collection of transversal matroids associated to its maximal tope linkage covectors by the combi…