Diamonds at $\omega_2$ under PFA

Question

Let $\lambda$ be a cardinal, and $S\subset \lambda^+$ be stationary. A $\diamondsuit^+(S)$-sequence is a sequence $\langle \mathcal{A}_\alpha\mid \alpha\in S\rangle$ such that each $\mathcal{A}_\alpha\subset P(\alpha)$ and $|\mathcal{A}_\alpha|\leq \lambda$, and for all $A\subset \lambda^+$, there is a club $C\subset \lambda^+$ such that for all $\alpha\in C\cap S$, $A\cap \alpha\in \mathcal{A}_\alpha$ and $C\cap \alpha\in \mathcal{A}_\alpha$.

I saw it mentioned that under PFA, $\diamondsuit^+(E^{\omega_2}_{\omega_1})$ holds, where $E^{\omega_2}_{\omega_1}=\{ \alpha\in \omega_2\mid \mathrm{cf}(\alpha)=\omega_1\}$, but I don't have the reference for this. I wonder if anyone has obtained more results about the diamonds at $\omega_2$ that hold under PFA, and if they are optimal in any sense.

Answer 1

See Theorem 7.13 of Baumgartner's paper ``Applications of the proper forcing axiom''. Indeed as it is stated there, we will need less to get the conclusion.

Theorem. Assume weak Kurepa hypothesis fails and $2^{\aleph_0}=\aleph_2$. Then $\Diamond^+(E^{\omega_2}_{\omega_1})$ holds.

Also note that $2^{\aleph_1}=\aleph_2$ implies $\Diamond(E^{\omega_2}_{\omega})$.