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 ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$ together with $2^{\lambda }={\lambda }^{+}$ implies ${\mathrm{♢}}_S$ 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 ${\text{NS}}_{{\lambda }^{+}}\upharpoonright S$ is non-saturated;
2. if $I[S;\lambda ]$ is fat and $2^{\lambda }={\lambda }^{+}$, then ${\mathrm{♢}}_S$ holds;
3. ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$ 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 ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$ fails, while $I[S;\lambda ]$ is fat for every stationary $S\subseteq {\lambda }^{+}$ that reflects stationarily often.

The stronger principle ${\mathrm{♢}}_{{\lambda }^{+}}^{\ast }$ 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.