This question is related to my question "[Forcing with c.c.c forcing notions, Cohen reals and Random reals](https://mathoverflow.net/questions/146800/forcing-with-c-c-c-forcing-notions-cohen-reals-and-random-reals)". 


A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds a minimal generic real.


I arrived to the idea of constructing such a forcing notion by considering the following:


1) Mathis forcing at $\omega$ is essentially the same as Prikry's forcing at some measurable cardinal,


2) Recently, a minimal Prikry type forcing at a measurable cardinal is constructed by Koepke-Räsch-Schlicht (see "A minimal Prikry-type forcing for singularizing a measurable cardinal, J. Symbolic Logic Volume 78, Issue 1 (2013), 85-100." and also [Is Prikry forcing minimal?](https://mathoverflow.net/questions/138694/is-prikry-forcing-minimal) ).


So a natural idea is to define a version of the above forcing for $\omega$. It turns out to me that such a forcing is constructed many years ago by Judah-Shelah in "Forcing minimal degree of constructibility" and there it is shown that under $CH$ we can choose nice ultrafilters so that the resulting forcing adds a minimal generic real which is minimal.


Let me describe the forcing I have in mind: Let $(A\_n: n<\omega)$ be a partition of $\omega$ into infinite sets and for each $n$ let $U\_n$ be a non-principal ultrafilter over $A\_n$. Let $\mathbb{P}$ consists of all pairs $(t, T)$ where $T \subseteq [\omega]^{<\omega}$
is a tree with trunk $t$ and for all $t \unlhd u\in T$ (where $\unlhd$ denotes end-extension relation) $Suc\_T(u) \in U\_{max(u)}.$ We define $(t, T) \leq (s, S)$ iff $T \subseteq S$ and $(t, T) \leq^\* (s, S)$ iff $T \subseteq S$ and $t=s.$ The following can be proved easily:


1) $(\mathbb{P}, \leq)$ satisfies the c.c.c.,


2) $(\mathbb{P}, \leq, \leq^\*)$ satisfies the Prikry property,


3) Let $G$ be $\mathbb{P}-$generic over $V$ and let $f\_G=\bigcup \{t: \exists T, (t, T)\in G \}.$ Then $f\_G$ is a real,


4) Suppose $d \subseteq f\_G$ is infinite, then $V[d]=V[f\_G]$ (the proof is the same as in the proof of Proposition 4.4 in Koepke-Räsch-Schlicht paper). 


**Question.** Is it consistent with $ZFC$ that there is no sequence of ultrafilters as above such that the corresponding forcing defined above adds a minimal generic real?


Note that in such a model $CH$ or $MA+\sim CH$ should fail.


**Remark 1.** If we can show that the forcing does not add a minimal generic real when all of our ultrafilters are on the same class of near-coherence ultrafilters, then the answer to question 1 will become positive, as it is consistent that there is just one class of near-coherence non-principal ultrafilters.


**Remark 2.** As pointed out by Dorais, if all of the ultrafilters are RK equivalent, then the resulting forcing does not add a minimal generic real.



A special c.c.c forcing notion and adding minimal generic reals
Groszek studied this and related forcings in *Combinatorics on ideals and forcing with trees* (JSL 52 (1987), 582–593; [MR0902978](http://www.ams.org/mathscinet-getitem?mr=902978)). (Note that Groszek allows each node $\sigma$ to have a different ideal, so yours is a special case of what Groszek denotes $L(\Sigma^+)$. Also note that $L(\Sigma^+) = L(\Sigma^1)$ when all ideals are maximal.)


Theorem 7 gives a sufficient condition — the simultaneous tree property — for the generic $L(\Sigma^+)$-real to have minimal degree. I'm not sure the exact strength of the existence of a $\Sigma$ with the simultaneous tree property is but it feels like it should be possible to construct such a thing under CH. (The paper contains a construction under V = L.)


