454 problems

Let $C\in\mathbb{C}^{d\times d}$ be a scaling matrix of rank four. A principal $4$-minor is non-vanishing when the corresponding principal submatrix has nonzero determinant. Rank-f…

Fix a code ${\mathcal M}$. Given a code ${\mathcal C}$ of length $n$, polynomial-time minor-containment conjecture. It is decidable in time polynomial in $n$ whether ${\mathcal C}$…

Aluffi–Chen–Marcolli conjecture. The polynomial $P_{\overline{\mathcal{M}}_{0,n+1}}(t)$ is real-rooted.

Let $M$ and $M'$ be non-isomorphic matroids, and let $Z_M$ and $Z_{M'}$ denote their matroidal cycles in their respective ambient vector spaces. Suppose there exists an invertible…

Path 3-colorability conjecture. Every path is matroidally $3$-colorable.

Alternating path covering conjecture. There exists a path that alternates between $A\setminus B$ and $B\setminus A$ and covers $\mathcal C[A\cup B]$.

Let $M$ be a matroid, and call a rank-$k$ flat ordinary if it is the direct sum of a point and a rank-$(k-1)$ flat. Ordinary-flat conjecture. For each integer $k\ge 2$, every compl…

Let $\mathsf{M}$ and $\mathsf{N}$ be matroids on the same ground set, and suppose there is a rank-preserving weak map $\mathsf{N}\to\mathsf{M}$. Let…

Let $\mathsf{M}$ be a loopless matroid of rank $k$ on a ground set of size $n$. Let $\mathsf{U}_{k,n}$ be the uniform matroid of rank $k$ on $n$ elements, and write…

Bonin's conjecture. The matroid $M$ has at most

Let $G=(V,E)$ be a rigid graph and let $u,v\in V$. A pair of vertices is globally linked if every generic realisation of $G$ determines the distance between its two vertices. Globa…

Disjoint-independent-transversal conjecture. If $m\ge2n-1$, then $A$ has $m-(n-1)$ pairwise disjoint ITs.

Disjoint-transversal conjecture. If $m\ge2n-1$, then $A$ has $m-(n-1)$ pairwise disjoint transversals.

Let ${\mathcal A}$ be a central hyperplane arrangement with connected underlying matroid, and let $C({\mathcal A})$ denote its complement. Homotopy-type conjecture. The homotopy ty…

Corrected Welsh conjecture. For every triple $(n,r,b)$ such that $0 \leq r \leq n$ and $1 \leq b \leq \binom{n}{r}$, there is a matroid of rank $r$ on $n$ elements with exactly $b$…

Let $L$ be an $(n-k)$-dimensional linear subspace of $K^n$ whose tropical Plücker coordinates are all finite, and let $M\in K^{m\times n}$, with $m\geq k$, have nonzero rows in the…

Let $\mathcal{T}_{n,d}$ be the matroid introduced from the combinatorial encoding of arrangements of $d$ generic complete flags in $\mathbb{C}^n$. Let $n\Delta_{d-1}$ be the Minkow…

Let $\mathcal{T}_{n,d}$ be the matroid introduced from the combinatorial encoding of arrangements of $d$ generic complete flags in $\mathbb{C}^n$. Let $n\Delta_{d-1}$ be the Minkow…

Combinatorial characterization conjecture. If the assignment $M\mapsto G_M$ satisfies 1. for every such decomposition,

Gamma-polynomial conjecture.

K-theoretic matroid invariant conjecture. For any matroids $M$, $M_1$, and $M_2$,

Let a matroid belong to one of the listed classes: planar-graph matroids, graph matroids, regular matroids, sixth-root-of-unity matroids, half-plane-property matroids, or B-Rayleig…

Let $Z(\omega;\mathbf y)$ be Rayleigh, let $\widetilde Z$ be its symmetrization, and let $z(e)$ and $z(ef)$ be the auxiliary quantities defined in the source. Pairwise symmetrizati…

Let $Z(\omega;\mathbf y)$ be Rayleigh, and let $\widetilde Z$ be its symmetrization. For each $e\in E$ and each pair $\{e,f\}\subseteq E$, use the auxiliary quantities $z(e)$ and…

Symmetrization conjecture. If $Z(\omega;\mathbf y)$ is Rayleigh, then $\widetilde Z(\omega;\mathbf y)$ is Rayleigh. This is stated as an equivalent form of the Big Conjecture and r…