Bounded Martin's Maximum, weak Erdõs cardinals, and ψ_AC

We prove that a form of the Erdos property (consistent with V = L[H-omega2] and strictly weaker than the weak Chang's Conjecture at omega(1)), together with Bounded Martin's Maximum implies that Woodin's principle psi(AC) holds, and therefore 2(N0) = N-2. We also prove that psi(AC) implies that every function f : omega(1) --> omega(1) is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum fails.