On the normally torsion-freeness of square-free monomial ideals

Let $I\subset R=K[x_1,\dots ,x_n]$ be a square-free monomial ideal, $\mathfrak{q}$ be a prime monomial ideal in $R$, $h$ be a square-free monomial in $R$ with $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(h)\cap (\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathfrak{q})\cup \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(I))=\mathrm{\varnothing }$, and $L:=I\cap (\mathfrak{q},h)$. In this paper, we first focus on the associated primes of powers of $L$ and explore the normally torsion-freeness of $L$. We also give an application on a comb inatorial result. Next, we study when a square-free monomial ideal is minimally not normally torsion-free. Particularly, we introduce a class of square-free monomial ideals, which are minimally not normally torsion-free.