On Cohen and Prikry Forcing Notions

Abstract

We show that it is possible to add ${\kappa }^{+}-$Cohen subsets to $\kappa$ with a Prikry forcing over $\kappa$. This answers a question from \cite{HayutBenhanouGitik}. 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 \cite{BenhamouGartieShelah}. A situation with extender-based Prikry forcings is examined. This relates to a question of H. Woodin.

Corrected typos, added details to the proof of the main theorems, and improved the result about the Merimovich Extender-based Prikry forcing: we now prove that in its general form, it cannot answer Woodin's question

References in corpus (3)

• Intermediate Models of Magidor-Radin Forcing- Part II
• Negating the Galvin Property
• Kurepa trees and the failure of the Galvin property

Cited by in corpus (3)

• Cofinal types of ultrafilters over measurable cardinals
• Non-Galvin Filters
• Galvin's property at large cardinals and an application to partition calculus