35 problems

Let $P$ be the base polytope of a polymatroid. A triangulation of $P$ is unimodular if all its simplices are unimodular with respect to the relevant lattice, and it is flag if ever…

Exponential-growth conjecture. The sequence $\lambda_n$ grows at least exponentially in $n$.

Extremal polymatroid $a$-reducedness conjecture. The polymatroid $f$ is actually $a$-reduced; consequently,

Singleton-ratio growth conjecture. The true growth of this supremum is significantly smaller than the doubly exponential upper bound

For a vector of nonnegative integers $(q_{ij})_{1\leq i<j\leq5}$, let $S\subseteq(\mathbb{P}^1)^5$ be an irreducible threefold and let $\lambda\in\mathbb{Q}_{\geq0}$. The coefficie…

Let a $5$-variable polymatroid be a polymatroid for which the two families of non-Shannon inequalities proved in Theorem … hold for polymatroids with an $n$-copy. An $n$-copy is th…

Facet-and-vertex count conjecture. In general, the number of facets of $\mathcal Q^*_2$ is $2n+14$, and the number of vertices is $2n^2+17$. The displayed values agree with the rep…

Castillo–Cid-Ruiz–Mohammadi–Montaño conjecture. The Möbius support of $\mathscr{P}$ is a generalized polymatroid, that is, a homogenization of it yields a polymatroid. The paper se…

Symmetric exchange conjecture. The toric ideal

Let $E$ be a finite set, let $\mathrm{P}_1$ and $\mathrm{P}_2$ be polymatroids on $E$, and let $B(\mathrm{P}_1)$ and $B(\mathrm{P}_2)$ denote their base polytopes. Suppose that … F…

Let $E$ be a finite set, let $\mathrm{M}$ be a loopless matroid on $E$, and let $\mathrm{P}$ be a polymatroid on $E$. For the line bundle $\underline{\mathcal{L}}_{B(\mathrm{P})}$…

Four-variable Ingleton conjecture. The sub-cone obtained by adding the Ingleton inequalities to the inequalities defining ${\cal H}(N)$ determines all probabilistic conditional-ind…

Entropy-ray conjecture. Every $h\in {\cal H}(N)$ is a positive multiple of some entropy function.

Rota's conjecture. For each field $\mathbb{k}$ of positive characteristic, there is a list of finite restrictions that characterizes the linear polymatroids over $\mathbb{k}$.

The hierarchical-order conjecture. If

Sticky polymatroid conjecture. A polymatroid is sticky if and only if every pair of its flats is modular; equivalently, any two extensions of the polymatroid have an amalgam if and…

Let $\mathcal A$ be an access structure. It is ideal if it can be realized by an ideal entropic polymatroid, equivalently by an ideal distribution scheme. Let $\mathcal A^\bot$ den…

Let $P$ be a participant set, let $\mathcal A\subset 2^P$ be an access structure, and let $\mathcal A^\bot$ denote its dual access structure. Let $\sigma(\mathcal A)$ denote the en…

Sticky matroid conjecture. No matroids other than flat-modular matroids are sticky; equivalently, if a matroid has a non-modular pair of flats, then it is not sticky. The conjectur…

Let $M$ be a connected $k$-polymatroid and let $N$ be a connected minor of $M$. A connected minor-preserving deletion or contraction is a deletion or contraction that remains conne…

Sticky polymatroid conjecture. Every sticky polymatroid is modular.

Finite-minimality conjecture. There are only finitely many $2$-polymatroids that are minimal with the property of being non-representable over $\mathbb F$ under these reduction ope…

Let a polymatroid be a finite set of points equipped with its linear relations, and associate to these relations the corresponding toric ideal. White's conjecture. All relations be…

Let $E$ be a finite set, let $\mu$ be a unitary connectivity function on $E$, and let $\mu\downarrow X$ denote the connectivity-function minor obtained by the operation defined in…

Let $E$ be a finite set, and let $u$ be the connectivity function on ${a,b,c,d}$ defined by … … and with all other values determined by symmetry. A connectivity function is unitar…