46 problems

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…

Wilmes' conjecture. For all $i\geq 1$, one has

Let $\Delta$ be a homology $(d-1)$-sphere over a field $\mathbf{k}$, with Stanley–Reisner ring $\mathbf{k}[\Delta]=S/I_\Delta$. An l.s.o.p. is a linear system of parameters…

Let $G$ be a graph with edge set $E(G)$, and let an ordering of $E(G)$ determine the monomial lower order ideal $L(G)$ and the ring $R(G)$. An ordering is said to produce an NBC ba…

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…

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.

Symmetric exchange conjecture. The toric ideal of the base ring of a discrete polymatroid is generated by symmetric exchange binomials.

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

Let $G$ be a forest, let $I(G)$ be its edge ideal, let $I(G)^{[k]}$ denote its $k$-th square-free power, let $\nu(G)$ be its matching number, and let $\mathrm{aim}(G,k)$ be the max…

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…

Let $K$ be a field, let $S=K[x_1,\ldots,x_n]$, and let $I\subset S$ be a squarefree monomial ideal. For a finitely generated graded $S$-module $M$, write $\operatorname{hdepth}(M)$…

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…