8 problems

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 $L_n^2$ 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…

For a $d$-uniform clutter $\mathcal{C}$, let $I(\mathcal{C})$ be its edge ideal, and let $\Omega_{d,k}$ consist of the clutters satisfying … … and having no proper induced sub-clut…

Let $G$ be a finite connected simple graph, let $K[G]$ denote its edge ring over a field $K$, and let $q\geq 3$. A graded ring has a $q$-linear resolution when its minimal graded f…

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…