34 problems

Cohen–Macaulayness conjecture. The complex $\Sigma_r(G)$ is Cohen–Macaulay if and only if $\operatorname{Con}_r(G)$ is co-chordal.

Let $\Gamma$ be a simplicial complex, and let $b_{i,j}(\Gamma)$ and $b^e_{i,j}(\Gamma)$ denote its symmetric and exterior iterated Betti numbers, respectively. The iterated Betti-n…

Let $\Gamma$ be a simplicial complex, let $\Delta(\Gamma)$ and $\Delta^e(\Gamma)$ denote its symmetric and exterior algebraic shifts, and let $I_{\Delta(\Gamma)}$ and…

Let $\mathcal W_n$ be the wheel graph on $n+1$ vertices, let $I(\mathcal W_n)$ be its edge ideal in $R=\mathbbk[x_1,\ldots,x_n,v]$, and let $\nu(\mathcal W_n)$ denote its matching…

The v-number bound. For every simple connected graph $G$,

Let $r,d\in\mathbb{Z}_{\ge 0}$, and let $G$ be a connected graph. Write $R$ for the polynomial ring associated with $G$, $J(G)$ for its cover ideal, $\operatorname{reg}(R/J(G))$ fo…

Let $G$ be a connected graph on $n$ vertices. Write $R$ for the polynomial ring associated with $G$, $J(G)$ for its cover ideal, $\operatorname{reg}(R/J(G))$ for the Castelnuovo–Mu…

Mascia–Rinaldo–Romeo conjecture. The following conditions are equivalent:

Hibi–Qureshi conjecture. The ideal $I_{\mathcal{P}}$ arises as the toric ideal of the edge ring of a finite simple graph if and only if $\mathcal{P}$ is simple.

Dominance threshold conjecture. As $d\to\infty$, for every fixed $p>0.1$, the probability that the random homogeneous ideal is dominant tends to $0$.

For an integer $q\geq 1$, let $\mathcal{S}_q=\mathsf{k}[x_A\colon \emptyset\neq A\subseteq [q]]$, and let $\mathcal{E}_q$ be the ideal generated by … For $r\geq 1$, write…

Let $I$ be a monomial ideal. A minimal generalized Lyubeznik resolution and a minimal generalized Barile–Macchia resolution are resolutions of $I$ with the indicated generalized co…

Anisotropy conjecture. For an arbitrary field $\mathbb{F}$, any $\mathbb{F}$-homology sphere is generically anisotropic over $\mathbb{F}$.

Let $\Delta$ be a simplicial tree of dimension $t-1$, let $I(\Delta)$ denote its facet ideal in the polynomial ring $R$, and let $\nu_1(\Delta)$ and $\nu(\Delta)$ denote its induce…

Let $T$ be a tree, let $t$ be a positive integer, and let $J_t(T)$ denote the $t$-connected ideal of $T$. Let $\operatorname{bight}(J_t(T))$ denote the big height of this ideal. Pr…

Let $T$ be a tree, let $t$ be a positive integer, and let $J_t(T)$ denote the $t$-connected ideal of $T$. Let $\mathcal H$ be the hypergraph corresponding to $J_t(T)$, and write…

Restricted bipartite connectivity conjecture. The stable value of the depth of the symbolic powers of $I(G)$ is

Let $I$ be a monomial ideal, let $\mathcal{F}_L$ be a Lyubeznik resolution, and let Algorithm $1$ be the trimming algorithm defined in the source. The resulting complex is compared…

Let $R$ be a polynomial ring over a field $\boldsymbol{k}$, and let $I\subseteq R$ be a monomial ideal. Let $\mathcal{F}_L$ be a Lyubeznik resolution of $R/I$. A Barile–Macchia res…

Let $G=(V,E)$ be a graph, let $n=|V|$, and let $k[G]=k[x_1,\ldots,x_n]/I(G)$ be its edge ring, where $I(G)$ is generated by $\{x_ix_j:\{i,j\}\in E\}$. Write…

Let $I$ be an Artinian monomial ideal and let $J$ be a polarization of $I$. Write $4\Delta(J)4$ for the simplicial complex associated to $J$. Ball-or-sphere conjecture. The simplic…

Let $G$ be a finite simple graph, let $J_G$ denote its binomial edge ideal, and let $C_n$ denote the cycle graph on $n$ vertices. Let $\mathcal{R}(J_G)$ be the Rees algebra of…

Billera–Lee upper-bound conjecture. The inequality

Let $4\mathfrak{C}_d4$ denote the class of chordal $d$-uniform clutters and let $4\mathfrak{C}'_d4$ denote the class of decomposable $d$-uniform clutters. The inclusion conjecture.…

Algebraic and symmetric shifting conjecture. For all $i,j$,