18 problems

Zanello's conjecture. If $R/I$ is level of codimension three, then

Let $\tau$ be a preorder and let $\tau^\ast$ denote its dual preorder. Preorder h-vector duality conjecture. One has … for every preorder $\tau$. The statement substantially genera…

Let $\tau$ be a preorder of size $n$. For $0\leq i\leq n$, let $h_i(\tau)$ count the lattice points of $\mathcal Q_\tau$ with exactly $i$ nonzero coordinates, and write…

Let $\tau$ be an arbor on the ground set $[n]:=\{1,2,\dots,n\}$, and let $\mathcal Q_\tau$ be its associated $n$-dimensional arbor polytope. Define…

The arbor polytope h-vector realization conjecture. For every arbor $t$, there exists a simple polytope $\Phi_t$ such that

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 $G$ be a cycle graph with $\ell\geq 3$, let $R=E_{\mathbb{K}}[\operatorname{STAB}(G)]$ be its Ehrhart ring, and write its h-vector as $(h_0,h_1,\ldots,h_s)$ with $h_s\neq 0$. S…

Let $\mathbb P^n$ be projective $n$-space, and let $\mathbb{S}(\ell_1,\ldots,\ell_r)$ and $\mathbb{S}(m_1,\ldots,m_s)$ denote contact star configurations formed from distinct hyper…

Let $s\le 4$ and let $m_i$ be the multiplicities of $s$ general fat points. For each $i$, take a contact star configuration formed by $m_i+1$ lines tangent to the same conic. The c…

Let $\Delta_{n,\ell}$ be the simplicial complex of $\ell$-zero-sumfree subsets, and let $h_1,\ldots,h_d$ be the relevant entries of its $h$-vector. A simplicial complex is pure whe…

Let $\Delta$ be a PS ear-decomposable simplicial complex, and let $h(\Delta)$ denote its $h$-vector. The PS ear-decomposability conjecture. The vector $h(\Delta)$ is a pure…

Characteristic-independence conjecture. An $h$-vector is level in some characteristic if and only if it is level in every characteristic.

Let $M$ be a loopless matroid, and let $(h_0,h_1,\ldots,h_s)$ be the $h$-vector of its broken circuit complex $BC(M)$, where $s$ is the largest index $j$ such that $h_j\ne0$. Flawl…

Let $H_{\mathrm{bal}}$ be the cone generated by the $h$-vectors of balanced $d$-polytopes or balanced homology $(d-1)$-spheres, and let $H_{\mathrm{flag}}$ be the cone generated by…

Cook–Nagel and Constantinescu–Varbaro conjecture. The following equality of sets holds:

A matroid complex is the simplicial complex of independent sets of a matroid. For a $(d-1)$-dimensional simplicial complex $\Delta$, its $h$-vector $h(\Delta)=(h_0,h_1,\ldots,h_d)$…

Let $h=(1,h_1,\dotsc,h_e)$ and $h'=(1,h'_1,\dotsc,h'_{e-1})$ be pure $O$-sequences. For a vector $H=(1,H_1,H_2,\dots,H_t)$, write $\Delta H=(1,H_1-1,H_2-H_1,\dots,H_t-H_{t-1})$ for…

A vector $h=(1,h_1,\dots,h_d) \in \mathbb{Z}^{d+1}$ is the $h$-vector of a $(d-1)$-dimensional Buchsbaum complex $\Delta$ such that $\beta_k(\Delta)=0$ for $k \leq d-3$. A vector…