116 problems

da Silva's conjecture. Every positively oriented matroid arises from a positroid in this way.

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 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…

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…

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

Clean root system conjecture. All rank $3$ root systems are clean.

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

Let $\mathcal{H}(d,n)$ be an arrangement of $n$ (pseudo)hyperplanes in $\mathbb{P}^d$, and call a maximal cell complete if it is bounded by every hyperplane of the arrangement. Rou…

Let an oriented matroid be a combinatorial analogue of a real vector space, and let an OM-Grassmannian be the finite simplicial complex associated analogously to the Grassmannian o…

Finiteness conjecture. There are only finitely many sporadic arrangements.

For positive integers $r$ and $n$, let $\mathcal{G}^{n,r}$ be the mutation graph of uniform oriented matroids of rank $r$ on $n$ elements. Cordovil–Las Vergnas conjecture. For all…

Ardila–Develin's correspondence conjecture. Tropical oriented matroids are in bijection with subdivisions of

Let a vector bundle be associated with a matroid bundle. MacPherson's conjecture. All characteristic classes of the vector bundle should be carried by the associated matroid bundle…

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 $C^n:=\{-1,1\}^n$ be the set of vertices of an $n$-dimensional cube over $\mathbb{R}$, and let the cube matroid be the matroid of the affine dependencies of $C^n$. An orientati…

Cryptomorphic reduction conjecture. These new representations should lead to orientation theories that can be cryptomorphically reduced to $D_n$-orientations.

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…

Existence conjecture. The class of wild orthogonally oriented matroids exists.

Let $\mathcal C$ be a neural code, and let $\mathbf{P}_{\mathbf{Code}}$ denote the poset of combinatorial neural codes. A code lies below another code when it is below it in this p…

A neural code is a subset of the power set $2^{[n]}$, where $[n]=\{1,\dots,n\}$. A code is convex if it has a realization by open convex subsets of some Euclidean space. Let…

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…

Uniqueness conjecture. For $k=1$, the reorientation class $[C_r(n)]$ is the unique reorientation class having $c_r(n,k)$ $k$-neighborly reorientations.

Let $\mathcal{A}=\{H_1,\dots,H_n\}$ be a real arrangement in $\mathbb{R}^\ell$. Choose $f_1,\dots,f_n\in\operatorname{\mathsf{N}}^2(\mathcal{A})$ forming a good generator of…