Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $c=(c_1, \ldots, c_n)$ of nonnegative integers, we introduce the ideal $I_c \subset S$ which is generated by those monomials $x_1^{a_1} \cdots x_n^{a_n}$ belonging to $I$ with each $a_i \leq c_i$. A fundamental observation is that, if $I$ is generated in one degree and has linear quotients, then $I_c$ has linear quotients. Let $\delta = \delta_c(I)$ denote the biggest integer $q$ for which $(I^q)_c \neq (0)$. We are especially interested in when $I$ is an edge ideal. Let $G$ be a finite graph on $[n]$ and $I(G) \subset S$ the edge ideal of $G$. It will be reported that $(I(G)^\delta)_c$ is a polymatroidal ideal. In particular, $(I(G)^\delta)_c$ has linear quotients.

This is a joint work with Seyed Amin Seyed Fakhari, arXiv:2502.01768.