Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic filters $\{ G\_r : r \in \mathbb{R} \}$?


If CH holds, then the answer is yes. Recursively on the countable ordinals, we construct models $M\_\alpha = M[G\_\alpha]$, where $G\_\alpha \subseteq \mathbb{P}$ meets all dense sets in $\bigcup\_{\beta<\alpha} M\_\beta$.


We can also construct $2^\omega$ many distinct generic extensions. We can build (externally) a tree $T \subseteq \mathbb{P}$, where $T = \{ p\_s : s \in 2^{<\omega} \}$. Make $T$ such that: **(a)** If $|s| = n$, then $p\_s \in D\_n$, where $\{ D\_n : n \in \omega \}$ lists the dense subsets of $M$. **(b)** For $s$ a prefix of $t$, $p\_s \geq p\_t$. **(c)** For all $s$, $p\_{s0} \perp p\_{s1}$. Every real $r$ determines a distinct branch through $T$ and an associated $\mathbb{P}$-generic filter $G\_r$. So we have $2^\omega$ many distinct filters, and since the models are countable, a pigeonhole argument shows that we must have $2^\omega$ distinct models. But this argument does not show that the filters are mutually generic.



continuum many mutually generic filters
I think you can handle this by slightly modifying the construction in the last paragraph of your question. In addition to what you did there, enumerate all the dense subsets of $\mathbb P\times\mathbb P$, say as $E\_n$ ($n\in\omega$). Do this in such a way that every dense open set occurs infinitely often in the enumeration. Then at the stage of your recursion where you handle nodes $s$ of length $n$, extend your $p\_s$'s so that every two of them, considered as a pair, are in $E\_n$. That's only finitely much additional work at stage $n$, because there are only $2^n$ $s$'s to take care of. At the end of the construction, if $G$ and $G'$ are two of the generic filters you've produced, corresponding to paths $r$ and $r'$ through $T$, these will be mutually generic because any dense subset of $\mathbb P\times\mathbb P$ occurs as $E\_n$ for infinitely many $n$, in particular for some $n$ beyond the level where $r$ and $r'$ branched apart.


