Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications

Let  $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. We say that $I$ satisfies the persistence property if  $mathrm{Ass}_R(R/I^k)subseteq mathrm{Ass}_R(R/I^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{Ass}_R(R/I)$ denotes the set of associated prime ideals of $I$. In this paper, we introduce a class of square-free monomial ideals in the polynomial ring  $R=K[x_1,ldots,x_n]$ over field $K$ which are associated to  unrooted trees  such that if  $G$ is a unrooted tree and $I_t(G)$ is the ideal generated by the paths of $G$ of length $t$, then $J_t(G):=I_t(G)^vee$, where $I^vee$ denotes the Alexander dual of $I$, satisfies the persistence property. We also present a class of graphs such that the path ideals generated by paths of length two satisfy the persistence property. We conclude  this paper by giving a criterion for normally torsion-freeness of monomial ideals.