The P-ideal dichotomy (PID) is one of the most important and strongest consequences of the Proper Forcing Axiom (PFA). It has been observed that under PID, several mathematical propositions (not necessarily from set theory) become equivalent to an assertion regarding cardinal invariants. Stevo Todorcevic and Dilip Raghavan introduced the following project:
Let p be a consequence of PFA. Is there a cardinal invariant j such that p and j > ω₁ are equivalent under PID?
We may wonder if there is such cardinal invariant for the Open Graph Axiom (OGA) or for the Baumgartner Axiom for ω₁-dense sets (BA(ω₁)). It turns out that the conjunction of P-ideal dichotomy and Martin Axiom (MA) does not decide OGA or BA(ω₁). In this talk, we will mention some aspects of the proof. This is a joint work with Stevo Todorcevic.