On the embedded associated primes of monomial ideals

Let $I$ be a square-free monomial ideal in polynomial ring $R=K[x_1,\ldots, x_n]$ over field $K$, $\mathfrak{m}=(x_1, \ldots, x_n)$ the graded maximal of $R$, and $\{u_1, u_{\beta_1(I)}\}$ independent set minimal generators such that $\mathfrak{m}\setminus x_i \notin \mathrm{Ass}(R/(I\setminus x_i)^t)$ for all $x_i\mid \prod_{i=1}^{\beta_1(I)}u_i$ some positive integer $t$, where $I\setminus x_i$ denotes deletion at $x_i$ $\beta_1(I)$ maximum cardinality an $I$. In this paper, we prove if $\mathfrak{m}\in \mathrm{Ass}(R/I^t)$, then $t\geq \beta_1(I)+1$. As application, verify under certain conditions, every unmixed K\"onig is normally torsion-free, so has strong persistence property. addition, show transversal polymatroidal torsion-free. Next, state results on corner-elements ideals. particular, $R=K[x_1, $K$ $z$ $I^t$-corner-element $t$ \mathrm{Ass}(I\setminus x_i)^t$ $1\leq i \leq n$, divides $z$.