8 problems

Let $G$ be a graph with edge ideal $I(G)$. Suppose that, for some integer $q\in\mathbb{N}$, $I(G)^q$ has linear quotients with an ordering $\mathcal N^{(q)}:u_1>\dots>u_r$ of its g…

Linear quotients conjecture for symbolic powers. If $I(G)$ has a linear resolution, then $I(G)^{(k)}$ has linear quotients for every $k\geq 1$.

Let $I\subset S$ be a componentwise polymatroidal ideal in a polynomial ring $S=K[x_1,\dots,x_n]$. Power conjecture. Every power $I^k$ of $I$ has linear quotients. The source gives…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$. A monomial ideal $I\subset S$ is componentwise polymatroidal when each component $I_{\langle j\rangle}$ is polymatro…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. For each $j\geq 0$, let $I_{\langle j\rangle}$ be the monomial ideal gener…

Simon’s conjecture for monomial ideals. If $G(I)\subset G(\mathfrak{m}^d)$, then $I$ can be extended to $\mathfrak{m}^d$ by linear quotients.

Lexicographic homological quotients conjecture. For all $k\geq 0$ and all $\ell\geq 1$, the ideal $\operatorname{HS}_k((I(G)^\vee)^\ell)$ has linear quotients with respect to the l…

Let $K$ be a field, let $S=K[x_1,x_2,\ldots,x_n]$, let $I\subset S$ be a proper monomial ideal with linear quotients, and write $[n]=\\{1,\ldots,n\\}$. The stronger variable reduct…