$\{A_\alpha:\alpha<\omega_1\}$ Prove that there exist stationary $S\subseteq\omega_1$ such that $\{A_\alpha:\alpha\in S\}$ is $\Delta$-system. Let $\{A_\alpha:\alpha<\omega_1\}$ be family of finite sets. Prove that there exist stationary $S\subseteq\omega_1$ such that $\{A_\alpha:\alpha\in S\}$ is $\Delta$-system.
I wanted to use induction here. But even if we assume that all sets have exactly one element, I do not know how to proceed. The only regressive function I have in mind is something like $f(\alpha)=x\in A_\alpha$ if $x<\alpha$ and $0$ otherwise (we can assume none of the sets contain $0$).
Now the problem is when $S=\{\alpha:f(\alpha)=0\}$ is a stationary set we have from Fodor lemma. All we can do here is construct $\Delta$-system with empty root. But how to ensure that we won't lose stationarity here?
 A: Claim. Let $f \colon \omega_1 \to \omega_1$ be a function. If $f$ is not constant on a stationary set $S \subseteq \omega_1$ then there is some club $C \subseteq \omega_1$ such that for all $\alpha < \beta$ in $C$
$$
f(\alpha) \neq f(\beta).
$$
Proof. By our assumption $f^{-1}\{\alpha\}$ is not stationary for all $\alpha < \omega_1$. In particular, for all $\alpha < \omega_1$ there is some club $C_{\alpha} \in \omega_1$ such that for all $\beta \in C_{\alpha}$
$$
f(\beta) \neq f(\alpha).
$$
Let $C := \Delta_{\alpha < \omega_1} C_{\alpha}$. $C$ is a club and for $\alpha < \beta$ in $C$ we have $\beta \in C_{\alpha}$ and hence
$$
f(\alpha) \neq f(\beta).
$$
Q.E.D.
Consider your sequence $\{ A_{\alpha} \mid \alpha < \omega \}$. Fix $S \subseteq \omega_1$ stationary and some $n < \omega_1$ such that for all $\alpha \in S$
$$
A_{\alpha} = \{a_{\alpha}^0 < \ldots < a_\alpha^n \}
$$
has size $n$. Consider, for $i \le n$
$$
f_i \colon S \to \omega_1, \alpha \mapsto \langle i,  \omega + a^i_\alpha \rangle,
$$
where $\langle \ , \ \rangle$ is Gödel's pairing function. If possible, let $T_i \subseteq S$ be stationary such that $f_i \restriction T_i$ is constant. In this case let $r_i$ be the unique ordinal $r$ s.t. $f_i \restriction T_i = \langle i, \omega + r \rangle$. Otherwise let $T_i \subseteq S$ be stationary such that for all $\alpha < \beta \in T_i \colon f_i(\alpha) \neq f_i(\beta)$ and $f(\alpha) \not \in \{ \langle(0, \omega + r_0 \rangle, \ldots, \langle i-1, r_{i-1} \rangle \}$. In this case we let $r_i := -1$. It's easy to see that
$$
\{ A_{\alpha} \mid \alpha \in \bigcap_{i \le n} T_i \}
$$
is a $\Delta$-system with root
$$
R := \{r_i \mid i \le n \wedge r_i \ge 0 \}.
$$
