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Strongest Transformations

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Abstract

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 \(\textsf {ZFC} \). Here we address the consistency of the strongest conceivable transformations. Along the way, we obtain new results on Shelah’s coloring principle \({{\,\textrm{Pr}\,}}_1\): For \(\kappa \) inaccessible, we prove the consistency of \({{\,\textrm{Pr}\,}}_1(\kappa ,\kappa ,\kappa ,\kappa )\). 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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Notes

  1. The definition of the cardinal characteristic \(\chi (\kappa )\) is reproduced in Definition 3.19 below.

  2. Recall Convention 2.1.

  3. \(\chi (\kappa )\) should be understood as a measure of how far \(\kappa \) is from being weakly compact. By [28, Theorem 6.3;.5] if \(\kappa \) is weakly compact, then \(\chi (\textbf{C})=1\) for every C-sequence \(\textbf{C}\) over \(\kappa \).

  4. Recall that by Propositions 2.10 and 2.11, this only makes sense for \(\kappa \) a limit cardinal.

  5. Here, \(\mu ^+\) stands for the successor of \(\mu \) in the generic extension.

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Acknowledgements

The first author is partially supported by the European Research Council (grant agreement ERC-2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18). The second author is supported by the Foreign Postdoctoral Fellowship Program of the Israel Academy of Sciences and Humanities and by the Israel Science Foundation (grant agreement 2066/18). The main results of this paper were presented by the first author at an online meeting of the Toronto Set Theory Seminar, February 2021. He thanks the organizers for the invitation and the participants for their feedback. The authors thank the anonymous referees X, Y and Z for their feedback. We are especially grateful to anonymous referee X for an exceptionally detailed and thorough report.

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  1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel

    Assaf Rinot & Jing Zhang

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  1. Assaf Rinot
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Correspondence to Assaf Rinot.

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Rinot, A., Zhang, J. Strongest Transformations. Combinatorica 43, 149–185 (2023). https://doi.org/10.1007/s00493-023-00011-0

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