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A Silver forcing "below $2^n$" is defined e.g. in Definition 7.4.11 of [Tomek Bartoszyński and Haim Judah, Set Theory: on the structure of the real line, A. K. Peters, Wellesley, 1995.]. It is called Infinitely equal forcing EE there. In the same book, in Lemma 7.4.15 the authors show that EE preserves p-points. However, the proof of this lemma seems to be not complete/correct: the choice of conditions $p^{n+1}$ is not clear, as needed extensions of $p^n$ for various $r_n^j$ may "contradict" each other. (This would not be a problem if conditions were $2^n$-splitting trees, i.e., if the forcing were more like the Sacks rather than Silver.)

Do you know another source/reference for the proof that EE preserves p-points? Or perhaps I am missing something and the proof in the book is actually complete?

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David Chodounský and Osvaldo Guzmán showed in arXiv:1703.02082, There are no P-points in Silver extensions that There are no P-points in Silver extensions. They prove that

after adding a Silver real no ultrafilter from the ground model can be extended to a P-point, and this remains to be the case in any further extension which has the Sacks property.

In particular, any free ultrafilter from the ground model will not be a p-point, and hence cannot be an ultrafilter, as Silver forcing is proper.

Silver is a complete subforcing of the forcing EE (right? In each entry of the generic, use the first bit — this will be a Silver generic) so also EE destroys all p-points.

(Edit 2022: Published 2019 in the Israel Journal of Mathematics, vol 232 (2) 759–773, https://doi.org/10.1007/s11856-019-1886-2)

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    $\begingroup$ So as everyone else is too polite to explicitly say it, I will, to avoid any confusion: As Martin points out, 7.4.15 in B-J is incorrect. $\endgroup$
    – Jakob
    Commented Jan 19, 2022 at 15:17

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