We study which [Formula: see text]-distributive forcing notions of size [Formula: see text] can be embedded into tree Prikry forcing notions with [Formula: see text]-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a [Formula: see text]-distributive forcing notion of size [Formula: see text] be extended to a [Formula: see text]-complete ultrafilter.

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. We study which [math]-distributive forcing notions of size [math] can be embedded into tree Prikry forcing notions with [math]-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a [math]-distributive forcing notion of size [math] be extended to a [math]-complete ultrafilter.