Joint work with Jing Zhang.
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.
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Jing and I recently found an extension of Theorem A, showing that also $\square(\mu^+)$ implies $Pl_2(\mu^+,E^{\mu^+}_\mu,\mu)$. In particular, if $Pr_1(\omega_2,\omega_2,\omega_2,\omega_1)$ fails, then $\omega_2$ is weakly compact in $L$. The result will appear in a separate paper.
Submitted to Combinatorica, April 2021.
Accepted, May 2022.