Strongest transformations

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 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.

 

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2 Responses to Strongest transformations

  1. saf says:

    Jing and I recently found an extension of Theorem A, showing that also ◻(μ+) implies Pl2(μ+,Eμμ+,μ). In particular, if Pr1(ω2,ω2,ω2,ω1) fails, then ω2 is weakly compact in L. The result will appear in a separate paper.

  2. saf says:

    Submitted to Combinatorica, April 2021.
    Accepted, May 2022.

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