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