We answer a question of Carroy, Levin and Notaro by constructing, for each n, a model in which there are n-entangled sets but not n+1-entangled sets(not even n+1-increasing). This is achieved by defining a parametrized version of PFA associated to an n-entangled set.

We also will show that under CH or after adding omega_1-many Cohen reals, there is a collection <X_n> of pairwise isometric sets of the cantor space such that, for each n, X_n is n-entangled but not n+1-entangled.