A regular space is called an S-space if every subspace is separable, but not every subspace is Lindelöf. Their existence is implied by CH and compatible with MA, but PFA implies there are no S-spaces. Much like other questions resolved differently by CH and PFA, two questions naturally arise: (1) how much of PFA is required to have no S-spaces, and (2) which parts of PFA are compatible with S-spaces?

A fruitful area of recent research to separate properties of PFA has been to consider a version of PFA `relativised' to the existence of some object that otherwise cannot exist under PFA. The most notable example is PFA(S), where we restrict PFA to only the proper forcings that preserve some Souslin tree S. Other examples include PFA relativised to a 2-entangled set of reals or a weak HFD space.

In this talk we will present generalised techniques for relativising PFA to various objects, including several different S-spaces. We will then use this to explore our two above questions, including providing a negative answer to Todorčević's question of whether the P-Ideal Dichotomy + b > \aleph_1 implies there are no S-spaces.