6 problems

Let $G$ be a graph, let $I(G)$ be its edge ideal, and let $I(G)^{[p]}$ denote the ideal generated by the squarefree monomials among the generators of $I(G)^p$. Write $\nu_0(G)$ for…

Let $G$ be a forest, let $I(G)$ be its edge ideal, and for $q\geq 1$ let $I(G)^{[q]}$ be the ideal generated by squarefree monomials corresponding to matchings of size $q$ in $G$.…

Let $C_n$ be a cycle on $n$ vertices, let $W(C_n)$ denote its whiskered cycle, and let $I(G)$ be the edge ideal of a finite simple graph $G$. For $q\geq 1$, let $I(G)^{[q]}$ be the…

Normalized depth conjecture. For any $I\subset S$, the function $g_I(k)$ is non-increasing.

Normalized-depth conjecture. The function $g_I(k)$ is nonincreasing.

Let $G$ be a graph, let $I(G)$ be its edge ideal, let $I(G)^{[k]}$ denote its $k$-th squarefree power, and let $\lambda(I(G))$ be the smallest number such that $I(G)^{[k]}$ is line…