The P-ideal dichotomy, Martin’s axiom and entangled sets

We build a model of the P -ideal ichotomy (PID) and Martin’s axiom for ω 1 ( MA_ω₁ MA ω 1 ) in which there is a 2-entangled set of reals. In particular, it follows that the Open Graph Axiom or Baumgartner’s axiom for ω 1 -dense sets are not consequences of PID + MA_ω₁ PID + MA ω 1 . We review Neeman’s iteration method using two type side conditions and provide an alternative proof for the preservation of properness.