Rigidity properties of Borel ideals on the integers

We classify the Borel ideals I on the set of natural numbers N for which p(N)I can be Borel embedded into the orbit space of a Borel action of the infinite symmetric group. As a consequence we show that certain Borel ideals I, including the Fréchet ideal, are completely characterized by the “Borel cardinality” of the set p(N)I.