The failure of diamond on a reflecting stationary set

Gitik, Moti; Rinot, Assaf

doi:10.1090/S0002-9947-2011-05355-9

The failure of diamond on a reflecting stationary set
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Trans. Amer. Math. Soc. 364 (2012), 1771-1795 Request permission

Abstract:

1. It is shown that the failure of $\diamondsuit _S$, for a set $S\subseteq \aleph _{\omega +1}$ that reflects stationarily often, is consistent with $\textsf {GCH}$ and $\mathrm {AP}_{\aleph _\omega }$, relative to the existence of a supercompact cardinal. By a theorem of Shelah, $\textsf {GCH}$ and $\square ^*_\lambda$ entails $\diamondsuit _S$ for any $S\subseteq \lambda ^+$ that reflects stationarily often.

2. We establish the consistency of existence of a stationary subset of $[\aleph _{\omega +1}]^\omega$ that cannot be thinned out to a stationary set on which the $\sup$-function is injective. This answers a question of König, Larson and Yoshinobu in the negative.

3. We prove that the failure of a diamond-like principle introduced by Džamonja and Shelah is equivalent to the failure of Shelah’s strong hypothesis.

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