Prikry-type forcings after collapsing a huge cardinal

TL;DR

This paper explores the effects of Prikry-type forcing on combinatorial principles after collapsing a huge cardinal, focusing on successors of singular cardinals and their associated ideals and partition relations.

Contribution

It extends the analysis of Prikry forcing effects from successors of regular cardinals to successors of singular cardinals, revealing preservation and destruction of certain ideal properties.

Findings

Prikry forcing preserves centeredness of ideals

Prikry forcing kills layeredness of ideals

Analyzes partition relations and transfer principles in the extension

Abstract

Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{\mu^{++}}{\mu^{+}}\to\binom{\kappa}{\mu^{+}}_{\mu}$ and $\mathrm{Tr}_{\mathrm{Chr}}(\mu^{+++},\mu^{+})$ in the extension by Prikry forcing at $\mu$.