**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:

- if $I[S;\lambda]$ is fat, then $\text{NS}_{\lambda^+}\restriction S$ is non-saturated;
- if $I[S;\lambda]$ is fat and $2^\lambda=\lambda^+$, then $\diamondsuit_S$ holds;
- $\square^*_\lambda$ implies that $I[S;\lambda]$ is fat for every $S\subseteq\lambda^+$ that reflects stationarily often;
- 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.

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An answer to the first part of Question 3 may be found in here.

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