Strongest Transformations

We continue our study of maps which transform high-dimensional complicated objects into squares of stationary sets. Previously, we proved that many such transformations exist in ZFC . Here we address the consistency of the strongest conceivable transformations. Along the way, we obtain new results on Shelah’s coloring principle Pr1 : For κ inaccessible, we prove the consistency of Pr1(κ,κ,κ,κ) . 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.