59 problems

Let $\mathcal{M}=(E,\mathcal{C})$ be an oriented matroid of rank $r$, let $e\in E$, and let $\mathcal{C}^+$ denote its positive circuits. An $e$-positive circuit is a positive circ…

Fix a realizable chirotope $\chi\in\{-1,1\}^{\binom{n}{3}}$. Let $\operatorname{Dr}^\chi_{3,n}$ be the chirotopal Dressian, consisting of the $\chi$-tropical Plücker vectors, and l…

Let $(E,\mathcal{L},g)$ be a simplicial affine oriented matroid, where $g$ is the distinguished element and $\mathcal{L}^{++}$ denotes its bounded complex. Simplicial bounded-compl…

Let $(E,\mathcal{L},g)$ be an affine oriented matroid, where $g$ is the distinguished element and $\mathcal{L}/g$ denotes contraction by $g$. The bounded complex is denoted by…

Let $\mathcal{M}$ be a realizable oriented matroid of rank $r$. Its extension space is the simplicial complex associated to the natural poset of one-point extensions of…

Let $\mathcal{A}$ and $\mathcal{B}$ be real central hyperplane arrangements, and let $R$ be an integral domain with $\operatorname{char}R\neq 2$. Write…

Let $S$ and $T$ be $d$-dimensional simplices whose interiors contain the origin. A vertex exchange replaces one vertex while producing another $d$-dimensional simplex whose interio…

Let $M$ be an oriented matroid of rank $r$, let $e$ be the distinguished element defining the embracing condition, and let $A$ and $B$ be bases of $M$. Call them $e$-embracing when…

$k$-Roudneff's conjecture. For any rank $r\geq 2k+1\geq 3$ oriented matroid $\mathcal{M}$ on $n\geq r+1$ elements,

Let $\mathcal{A}=\{H_1,\dots,H_n\}$ be a real arrangement in $\mathbb{R}^\ell$. A generalized tope graph $\operatorname{\widetilde{\mathcal{T}}}(y_1,\dots,y_n)$ is the graph constr…

Assume $\operatorname{char} R\neq 2$. Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be real arrangements. Their filtered VG algebras are $\operatorname{\mathcal{VG}}(\mathcal{A}_i)$, and…

A generic $(n,d)$-tropical oriented matroid is a tropical oriented matroid whose associated structures are generic, and an $(n,d)$-matching ensemble is the matching-field encoding…

Let $n\geq 3$, and choose uniformly at random a planar realizable chirotope $\chi$ of size $n$. For each realization of $\chi$, count the realizable one-point extensions, and avera…

For a finite point configuration $P\subset\mathbb{R}^d$ and an integer $k\geq 0$, let $[P]_k$ be the set of point configurations $Q\in(\mathbb{R}^d)^X$ such that every chirotope of…

Submodular-cut extension conjecture. There exists an extension $\mathcal{O}\cup p'$ corresponding to $\mathcal{M}'$.

Mandel's conjecture. Every oriented matroid $\mathcal{O}$ has an extension $g$ in general position (not a coloop) such that $(\mathcal{O}\cup g,g,f)$ is a Euclidean oriented matroi…

Let $M$ be an oriented regular matroid, equipped with triangulating circuit and cocircuit signatures. A consistent sandpile torsor algorithm is a consistent algorithm assigning san…

Braid-matroid equivariant injectivity conjecture. For all $i eggeq 1$, there exist equivariant injections

Let $\nu(r,k)$ be the largest $n$ such that every uniform oriented matroid of rank $r$ on $n$ elements has a $k$-neighborly reorientation. McMullen's conjecture. For every…

Let $H(d,n)$ be an arrangement of $n$ pseudohyperplanes in the real projective space $\mathbb{P}^d$, and call a $d$-cell complete if it is bounded by every pseudohyperplane. For an…

Let an $r$-signotope be a signotope of rank $r$, and call it 2-extendable if every pair of prescribed crossing points can be contained in an extending element of an extension to an…

A positive coline of a matroid is a coline whose copoint partition has more singular classes than multiple classes. A gammoid is a matroid arising from a directed graph by the gamm…

Let $M$ be a loopless oriented matroid with no $M(K_4)$ minor. A Hadwiger conjecture for oriented matroids. $M$ has a nowhere-zero $3$-coflow. This is the first non-trivial open ca…

Sign-vector reconstruction conjecture. The $(i-1)$-skeleton of $\partial P$ together with $\mathcal V_i(P)$ determines the combinatorial type of $P$, equivalently the entire abstra…

Forbidden-minor conjecture. A regular oriented matroid $M$ is non-even if and only if none of its $\mathsf{GB}$-minors is isomorphic to $M^*(\vec{K}_{m,n})$ for some $m,n\geq 2$ su…