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We study which κ-distributive forcing notions of size κ can be embedded into tree Prikry forcing notions with κ-complete ultrafilters under various large cardinal assumptions. An alternative formulation -can the filter of dense open subsets of a κ-distributive forcing notion of size κ be extended to a κ-complete ultrafilter. Definition 2. Let πWith the separative order p ≤ P/H q if an only if for every q ≤ P r, r is compatible with p.Claim 3. Let P, Q be any forcing notions, then:(1) Let G ⊆ P be V -generic and π :

In this paper, we answer a question asked in [14] 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.

We prove that Galvin's property consistently fails at successors of strong limit singular cardinals. We also prove the consistency of this property failing at every successor of a singular cardinal. In addition, the paper analyzes the effect of Prikry-type forcings on the strong failure of the Galvin property, and explores stronger forms of this property in the context of large cardinals. Contents 0. Introduction 1. Preliminaries 1.1. Notations 1.2. Clubs and Filters 1.3. Forcing preliminaries 1.4. Radin forcing with interleaved collapses 2. The consistency of the local and global failure 2.1. Local failure of Galvin's property 2.2. Global failure of Galvin's property 2.3. The impossibility of the ultimate global failure 2.4. Galvin's number at successor of singular cardinals 3. The strong failure and Prikry-type generic extensions 4. Stronger forms of Galvin's property on normal filters 5. Open problems 5.1. The failure at successors of singulars 5.2. Galvin's property and large cardinals 5.3. Consistency strength of the failure of Galvin's property 2010 Mathematics Subject Classification. 03E35, 03E55.