Can you have many independent reals? Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that:


*

*$\Bbb P_\alpha$ is c.c.c.

*$\Bbb P_\alpha$ adds a real which determines the generic.

*For every countable $A\subseteq\omega_1$, and $\alpha\notin A$ the finite support product $\prod_{\beta\in A}\Bbb P_\beta$ does not add a $V$-generic real for $\Bbb P_\alpha$?

 A: First add $\omega_1$ Cohen reals, then partition this set of Cohen reals into $\omega_1$ disjoint sets $A_i$ each of size $\omega_1$. Let $P_i$ be a sigma centered forcing whose generic is a real coding a meager set covering $A_i$. Then, it is easy to check that the family $\{P_i : i < \omega_1\}$ is as required.
Some details: Let $p \in Q_i$ iff $p = (F, \overline{n} = \langle n_k : k \leq N \rangle, \overline{\sigma} = \langle \sigma_k : k < N \rangle)$ where $F$ is a finite subset of $A_i$, $n_0 = 0, n_k \in \omega$ are increasing and $\sigma_k \in 2^{[n_k, n_{k+1})}$, $q \leq p$ iff $F_p \subseteq F_q$, $\overline{\sigma}_q, \overline{n}_q$ extend $\overline{\sigma}_p, \overline{n}_p$, and for every $x \in F_p$, every new string $\sigma$ from $\overline{\sigma}_q$, $x$ disagrees with $\sigma$ somewhere. $Q_i$ is sigma centered and adds a real, namely $\bigcup \{\overline{\sigma}_p : p \in G_{Q_i}\}$, coding a meager set covering $A_i$. Let $P_i$ be the complete subalgebra of $Q_i$ generated by this real. Now note that if $y$ is any real in $\prod \{P_i : i \neq i_0\}$, then the meager set coded by $y$ cannot cover $A_{i_0}$ - in fact none of the reals in $A_{i_0}$ is covered. Clauses 1 and 2 are obvious.
