A relative of the approachability ideal, diamond and non-saturation

Abstract: Let $\lambda$ denote a singular cardinal.
Zeman, improving a previous result of Shelah, proved that $\square^*_\lambda$ together with $2^\lambda=\lambda^+$ implies $\diamondsuit_S$ for every $S\subseteq\lambda^+$ that reflects stationarily often.

In this paper, for a subset $S\subset\lambda^+$, a normal subideal of the weak approachability ideal is introduced, and denoted by $I[S;\lambda]$. We say that the ideal is fat if it contains a stationary set. It is proved:

1. if $I[S;\lambda]$ is fat, then $\text{NS}_{\lambda^+}\restriction S$ is non-saturated;
2. if $I[S;\lambda]$ is fat and $2^\lambda=\lambda^+$, then $\diamondsuit_S$ holds;
3. $\square^*_\lambda$ implies that $I[S;\lambda]$ is fat for every $S\subseteq\lambda^+$ that reflects stationarily often;
4. it is relatively consistent with the existence of a supercompact cardinal
   that $\square^*_{\lambda}$ fails, while $I[S;\lambda]$ is fat for every stationary $S\subseteq\lambda^+$ that reflects stationarily often.

The stronger principle $\diamondsuit^*_{\lambda^+}$ is studied as well.

Updates:

In a subsequent paper, it is established that the hypothessis of the above theorem is optimal.

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Citation information:

A. Rinot, A relative of the approachability ideal, diamond and non-saturation, J. Symbolic Logic, 75(3): 1035-1065, 2010.