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The independence of \(\mathsf {GCH}\) and a combinatorial principle related to Banach–Mazur games

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Abstract

It was proved recently that Telgársky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of \(\mathsf {GCH}+\square \). The proof introduces a combinatorial principle that is shown to follow from \(\mathsf {GCH}+\square \), namely:

\(\bigtriangledown \)::

Every separative poset \({\mathbb {P}}\) with the \(\kappa \)-cc contains a dense sub-poset \({\mathbb {D}}\) such that \(|\{ q \in {\mathbb {D}} \,:\, p \text { extends } q \}| < \kappa \) for every \(p \in {\mathbb {P}}\).

We prove this principle is independent of \(\mathsf {GCH}\) and \(\mathsf {CH}\), in the sense that \(\bigtriangledown \) does not imply \(\mathsf {CH}\), and \(\mathsf {GCH}\) does not imply \(\bigtriangledown \) assuming the consistency of a huge cardinal. We also consider the more specific question of whether \(\bigtriangledown \) holds with \({\mathbb {P}}\) equal to the weight-\(\aleph _\omega \) measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of \(\mathsf {ZFC}+\mathsf {GCH}\).

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Correspondence to Will Brian.

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Brian, W., Dow, A. & Shelah, S. The independence of \(\mathsf {GCH}\) and a combinatorial principle related to Banach–Mazur games. Arch. Math. Logic 61, 1–17 (2022). https://doi.org/10.1007/s00153-021-00770-x

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  • DOI: https://doi.org/10.1007/s00153-021-00770-x

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