I'm trying to see that assuming $\diamondsuit$ the following holds:
- Exists $\{A_\alpha\}_{\alpha<\omega_1}$ such that $A_\alpha\subset\alpha\times\alpha$ and for every $A\subset \omega_1\times\omega_1$ the set $\{\alpha\in\omega_1: \alpha\times\alpha\cap A=A_\alpha\}$ is stationary.
- Exists $\{f_\alpha\}_{\alpha<\omega_1}$ such that $f_\alpha:\alpha\rightarrow \alpha$ and for every $f:\omega_1\rightarrow\omega_1$ the set $\{\alpha\in\omega_1: f\upharpoonright \alpha=f_\alpha\}$ is stationary.
Is anybody willing to give me a proof of this two facts?
Thanks in advance.
HINT: For the first one let $h:\omega_1\times\omega_1\to\omega_1$ be a bijection. Let $C=\{\alpha\in\omega_1:h[\alpha\times\alpha]=\alpha\}$, and show that $C$ is a club set in $\omega_1$. Then consider the sets $h^{-1}[A_\alpha]$ for $\alpha\in C$.
For the second, let $h$ and $C$ be as above. For $\alpha\in C$ let
$$f_\alpha=\begin{cases} \{\langle\xi,\eta\rangle\in\omega_1\times\omega_1:h(\langle\xi,\eta\rangle)\in A_\alpha\},&\text{if this is a function}\\ \varnothing,&\text{otherwise}\;, \end{cases}$$
and for $\alpha\in\omega_1\setminus C$ let $f_\alpha=\varnothing$.