16 problems

Eventual homological linearity conjecture. If $I(G)$ has a linear resolution, then $\operatorname{HS}_i(I(G)^k)$ has a linear resolution for all $i\geq 0$ and all $k\gg 0$.

Let $S$ be a polynomial ring and let $I\subset S$ be a monomial ideal. For every monomial prime ideal $P\subset S$, let $I(P)$ denote the monomial localization of $I$ at $P$, obtai…

Let $G$ be a finite simple graph with vertex set $V(G)=\{1,\ldots,n\}$, let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$, and let $I(G)\subseteq S$ be the edge ideal…

Let $G$ be a finite connected simple graph, let $\mathbb k[G]$ denote its toric edge ring, and let $q$ be a positive integer. A graded ring has a $q$-linear resolution when its min…

Let $L_n^2$ be the graph on $[1,n]$ with edges $\{i,i+1\}$ for $i=1,\ldots,n-1$ and $\{i,i+2\}$ for $i=1,\ldots,n-2$. Let $\Delta_k^t(L_n^2)$ be the associated complex, and let its…

Let $L_n$ be the path graph on vertices $\{x_1,\ldots,x_n\}$, and let $\Delta_k^t(L_n)$ denote the associated complex. The preceding result establishes the claim when $n=2k-1$. Lin…

The v-number conjecture. If $I$ has linear powers, then

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 $G$ be a finite connected simple graph, and let $K[G]$ denote its edge ring. For an integer $q\geq 4$, assume that $K[G]$ has a $q$-linear resolution. Hibi–Matsuda–Tsuchiya con…

Let $I(n,p)$ be the edge ideal of the Erdős–Rényi random graph $G(n,p)$. Assume that the remaining parameter regime satisfies $n^5(1-p)^8p^2\to\infty$, after excluding the regimes…

Let $G$ be a finite connected simple graph, and let $K[G]$ denote its edge ring. A graded ring has a $q$-linear resolution when its minimal graded free resolution is $q$-linear, an…

Strict-increase conjecture. If $\index(I)>1$, then

Let $I$ be a monomial ideal, and let its monomial localizations be the ideals obtained by localizing at monomial prime ideals. A monomial ideal is polymatroidal if its minimal mono…

Let $S=K[x_1,\ldots,x_n]$. A monomial ideal $I\subset S$ is polymatroidal if its minimal monomial generators are indexed by the bases of a polymatroid. For each monomial prime idea…

Let $G$ be a graph, and let $I(G)$ denote its edge ideal. Say that the complement of $G$ is $C_4$-free when it contains no induced cycle of length four. Peeva–Nevo conjecture. The…

Let $I(G)$ be the edge ideal of a graph $G$, and suppose that $I(G)$ is the example constructed by Peeva and Nevo, with Castelnuovo–Mumford regularity four. Peeva–Nevo conjecture.…