# The failure of diamond on a reflecting stationary set

Joint work with Moti Gitik.

Abstract:

1. It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal.
This should be comapred with a theorem of Shelah, that GCH and $\square^*_\lambda$ entails $\diamondsuit_S$ for any subset $S\subseteq\lambda^+$ 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 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.