This is probably well known but I'll appreciate pointers to references: Is there any model where for a singular cardinal $\kappa$ of cofinality $\omega$, Very Weak Square holds at $\kappa$ but every scale on $\kappa$ has stationarily many bad points of cofinality $\omega\_1$? 


Definitions: Very Weak Square at $\kappa$: there is a sequence $\langle C\_\alpha: \alpha<\kappa^+\rangle$ where for a club of $\alpha\in \kappa^+$ 


1. $C\_\alpha\subset \alpha$ is unbounded
2. for any bounded $x\in [C\_\alpha]^\omega$, there is $\beta<\alpha$ such that $x=C\_\beta$.


Scales at $\kappa:$ For some $\langle \kappa\_i: i\in \omega\rangle$ increasing cofinal regular cardinals, $\langle f\_\alpha\in \prod\_{i\in \omega} \kappa\_i: \alpha<\kappa^+\rangle$ is a scale if it is increasing and cofinal in $<^\*$ (increasing mod finite). $\alpha<\kappa^+$ is a good point if $cf(\alpha)>\omega$ and there is $A\subset \alpha$ cofinal and $m\in \omega$ such that $\{f\_\alpha\restriction\_{>m}: \alpha\in A\rangle$ is pointwise increasing. $\alpha$ is bad if it's not good.


It is known that the assertion is false at $\aleph\_\omega$ (Foreman-Magidor) as VWS at $\aleph\_\omega$ implies there is a club $C\subset \aleph\_{\omega+1}$ such that all $\alpha\in C\cap cof(\omega\_1)$ is good. But the proof goes through an intermediate principle (approachability) and VWS does not in general imply AP at larger singular cardinals.



Very weak square and good points
Though Very Weak Square at $\kappa$ does not in general imply $\mathrm{AP}\_\kappa$, it *does* imply that $\kappa^+ \cap \mathrm{cof}(\omega\_1)$ is in the approachability ideal on $\kappa^+$. In fact, the Very Weak Square sequence $\langle C\_\alpha \mid \alpha < \kappa^+ \rangle$ is itself a witness to this, provided that we have $\mathrm{otp}(C\_\alpha) = \omega\_1$ for all $\alpha \in \kappa^+ \cap \mathrm{cof}(\omega\_1)$, which can always be arranged. It then follows from the usual arguments, as in the Foreman-Magidor paper on Very Weak Square, that, in a scale on $\kappa$, the set of bad points of cofinality $\omega\_1$ is non-stationary.


