Strongest transformations

TL;DR

This paper investigates the consistency of the strongest transformations of high-dimensional objects into stationary sets, extending previous results and exploring implications for Shelah's coloring principle and Galvin's theorem.

Contribution

It establishes the consistency of the strongest transformations and provides new results on Shelah's coloring principle and the limitations of lifting Galvin's theorem.

Findings

Proves the consistency of Pr_1(,,,) for inaccessible .

Achieves a full lifting of Galvin's theorem for successors of regulars.

Shows the inconsistency of lifting Galvin's theorem to successors of singulars.

Abstract

We continue our study of maps transforming high-dimensional complicated objects into squares of stationary sets. Previously, we proved that many such transformations exist in ZFC, and here we address the consistency of the strongest conceivable transformations. Along the way, we obtain new results on Shelah's coloring principle $Pr_1$. For $\kappa$ inaccessible, we prove the consistency of $Pr_1(\kappa,\kappa,\kappa,\kappa)$. For successors of regulars, we obtain a full lifting of Galvin's 1980 theorem. In contrast, the full lifting of Galvin's theorem to successors of singulars is shown to be inconsistent.