It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model P-points (or Ramsey ultrafilters) are preserved by Sacks forcing (i.e. still generate an ultrafilter in the extension). Thus, the only examples of P-points or Ramsey ultrafilters that I know of in Sacks model are those that already existed in the ground model, hence they all have weight $\aleph\_1$ (i.e. they are generated by $\aleph\_1$ many elements, since the ground model satisfies CH). So my question is:


**Is it known whether there are P-points and/or Ramsey ultrafilters of weight $\aleph\_2$ in Sacks model?** (actually my real question is: if yes, how are they constructed?, I guess)



Ultrafilters of weight $\aleph_2$ in Sacks model
I assume that by "the Sacks model" you mean the result of a countable-support iteration of Sacks forcing for $\omega\_2$ steps over a ground model satisfying GCH. On that understanding, think the answer to your question is no. If $U$ is a P-point in the final model, then, by a reflection argument, there will be ordinals $\alpha<\omega\_2$ such that $U\cap V[G\_\alpha]$ is a P-point in $V[G\_\alpha]$ (where $G\_\alpha$ is the part of the generic ultrafilter that was produced by the first $\alpha$ stages of the iteration). Since $V[G\_\alpha]$ satisfies CH, this P-point $U\cap V[G\_\alpha]$ is generated by $\aleph\_1$ sets. Furthermore, because it's a P-point, it will be preserved through the later stages (from $\alpha$ to $\omega\_2$) of the iteration. That means that the $\aleph\_1$ generators of $U\cap V[G\_\alpha]$ actually generate all of $U$ in the final model.


