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We continue [4] and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.

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In this paper, we answer a question asked in Koepke et al. regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems.

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We develop the theory of cofinal types of ultrafilters over measurable cardinals and establish its connections to Galvin’s property. We generalize fundamental results from the countable to the uncountable, but often in surprisingly strengthened forms, and present models with varying structures of the cofinal types of ultrafilters over measurable cardinals.

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We address the question of consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions [Benhamou and Gitik, Ann. Pure Appl. Logic 173 (2022) 103107; Questions 7.8, 7.9], [Benhamou et al., J. Lond. Math. Soc. 108(1) (2023) 190–237; Question 5] and improve theorem [Benhamou et al., J. Lond. Math. Soc. 108(1) (2023) 190–237; Theorem 2.3].

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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.

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Abstract(1)We show that it is possible to add $\kappa ^+$ -Cohen subsets to $\kappa $ with a Prikry forcing over $\kappa $. This answers a question from [9].(2)A strengthening of non-Galvin property is introduced. It is shown to be consistent using a single measurable cardinal which improves a previous result by S. Garti, S. Shelah, and the first author [5].(3)A situation with Extender-based Prikry forcings is examined. This relates to a question of H. Woodin.

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Journal of Mathematical Logic, Volume 25, Issue 02, August 2025. We address the question of consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions [Benhamou and Gitik, Ann. Pure Appl. Logic 173 (2022) 103107; Questions 7.8, 7.9], [Benhamou et al., J. Lond. Math. Soc. 108(1) (2023) 190–237; Question 5] and improve theorem [Benhamou et al., J. Lond. Math. Soc. 108(1) (2023) 190–237; Theorem 2.3].

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In this paper, we improve Galvin’s Theorem for ultrafilters which are [Formula: see text]-point limits of [Formula: see text]-points. This implies that in all the canonical inner models up to a superstrong cardinal, every [Formula: see text]-complete ultrafilter over a measurable cardinal [Formula: see text] satisfies the Galvin property. On the other hand, we prove that supercompact cardinals always carry non-Galvin [Formula: see text]-complete ultrafilters. Finally, we prove that [Formula: see text] implies the existence of a [Formula: see text]-complete filter (...) which extends the club filter and fails to satisfy the Galvin property. This answers questions [8, Question 5.22], [4, Question 3.4] and questions, [7, Question 4.5], [6, Question 2.26]. (shrink)

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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.

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