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 \(\mathsf {ZFC}+\mathsf {GCH}\).

最近证明，与Banach-Mazur博弈中的部分信息策略有关的Telgársky猜想在\（\ mathsf {GCH} + \ square \）模型中失败。证明引入了从\（\ mathsf {GCH} + \ square \）遵循的组合原理，即：

\（\ bigtriangledown \） ::

每个带有\（\ kappa \）- cc的分离姿态\\ {{\ mathbb {P}} \\}都包含一个密集的子姿态\\ {{\ mathbb {D}} \}，这样\（| \ {q \在{\ mathbb {D}} \，：\中，对于每个\（p \ in {\ mathbb {P}} \）中的p \ text {扩展} q \} | <\ kappa \）。

我们证明该原理独立于\（\ mathsf {GCH} \）和\（\ mathsf {CH} \），从某种意义上说，\（\ bigtriangledown \）并不意味着\（\ mathsf {CH} \），和\（\ mathsf {GCH} \）并不意味着\（\ bigtriangledown \）假设一个巨大的基数是一致的。我们还考虑一个更具体的问题，即\（\ bigtriangledown \）是否与\（{\ mathbb {P}} \）等于权重- \（\ aleph _ \ omega \）度量代数相等。再次假设一个大基数的一致性，我们证明这个问题的答案独立于\（\ mathsf {ZFC} + \ mathsf {GCH} \）。