It is wellknown that, if $p$ is a Ramsey (selective) ultrafilter on $\omega$, then after adding a Sacks real $p$ remains an ultrafilter (well, it's really the upwards closure of $p$ the one that's an ultrafilter in the extension, but this is how they usually phrase it). The same result still holds if $p$ is "only" a Ppoint. My question is whether any similar results are known to hold for certain families of ultrafilters that aren't necessarily Ppoints. In particular, if anyone is aware of any result of the form "whenever $p$ is not a Ppoint then it is destroyed by Sacks forcing", I would be very interested in knowing. Many thanks!
1 Answer
Olga Yiparaki's 1994 thesis from the University of Michigan "On some tree partitions" characterizes the ultrafilters that are preserved by Sacks forcing as those that are "Ramsey" with respect to colorings related to the the HalpernLauchli Theorem. (She calls them "hltultrafilters".)
This class of ultrafilters is broader than the class of Ppoints, as the hltultrafilters are closed under finite products, see, for example, Proposition 2 of Miller's paper Ultrafilters with property (s).

$\begingroup$ That's quite neat... an actual characterization for ultrafilters preserved by Sacks forcing! Thanks for answering my question. $\endgroup$ May 5, 2014 at 22:28

$\begingroup$ It might be interesting to see if every hltultrafilter is RKabove a Ppoint. Andreas Blass may know the answer to such questions! $\endgroup$ May 7, 2014 at 2:26

$\begingroup$ Actually, my interest is to know whether union ultrafilters (which I know you have worked with in the past) are preserved under Sacks forcing. Union ultrafilters are certainly RKabove a Ppoint (actually above two nonnearcoherent ones, the images under max and min), but as of now I have no idea of whether they are hlt or not (or if some of them are and others aren't). $\endgroup$ May 7, 2014 at 12:20