On the strong persistence property and normally torsion-freeness of square-free monomial ideals

In this paper, we first show that any square-free monomial ideal in $K[x_1,x_2,x_3,x_4,x_5]$ has the strong persistence property. Next we will provide a criterion for a minimal counterexample to the Conforti-Cornuejols conjecture. Finally we give a necessary and sufficient condition to determine the normally torsion-freeness of a linear combination of two normally torsion-free square-free monomial ideals.