Strongest Transformations

Assaf Rinot; Jing Zhang

doi:10.1007/s00493-023-00011-0

Strongest Transformations

Assaf Rinot, Jing Zhang

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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 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.

Original language	English
Pages (from-to)	149-185
Number of pages	37
Journal	Combinatorica
Volume	43
Issue number	1
DOIs	https://doi.org/10.1007/s00493-023-00011-0
State	Published - Feb 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature.

Funding

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.

Funders	Funder number
European Commission	ERC-2018-StG 802756
Israel Academy of Sciences and Humanities
Israel Science Foundation	2066/18

Keywords

Proxy principle
Square
Stick
Strong colorings
Transformations of the transfinite plane
Walks on ordinals
xbox

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10.1007/s00493-023-00011-0

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Strongest Transformations

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