The failure of diamond on a reflecting stationary set

Joint work with Moti Gitik.

Abstract:

1. It is shown that the failure of ${\mathrm{♢}}_S$, for a subset $S\subseteq {\mathrm{\aleph }}_{\omega +1}$ that reflects stationarily often, is consistent with GCH and ${\text{AP}}_{{\mathrm{\aleph }}_{\omega }}$, relatively to the existence of a supercompact cardinal.
   This should be comapred with a theorem of Shelah, that GCH and ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$ entails ${\mathrm{♢}}_S$ for any subset $S\subseteq {\lambda }^{+}$ that reflects stationarily often.
2. We establish the consistency of existence of a stationary subset of $[{\mathrm{\aleph }}_{\omega +1}{]}^{\omega }$ that cannot be thinned out to a stationary set on which the sup-function is injective.
   This answers a question of Konig, Larson and Yoshinobu, in the negative.
3. We prove that the failure of a diamond-like principle introduced by Dzamonja and Shelah is equivalent to the failure of Shelah’s strong hypothesis.

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Citation information:

M. Gitik and A. Rinot, The failure of diamond on a reflecting stationary set, Trans. Amer. Math. Soc., 364(4): 1771-1795, 2012.