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:


1. $\Bbb P\_\alpha$ is c.c.c.
2. $\Bbb P\_\alpha$ adds a real which determines the generic.
3. 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 $\Bbb P\_\alpha$?



Can you have many independent reals?
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.


