PCF theory and good points in scales If $\kappa$ is a singular cardinal, a scale for $\kappa$ consists of an increasing sequence $\langle \kappa_i : i < \mathrm{cf}(\kappa) \rangle$ converging to $\kappa$ and a sequence of functions $\langle f_\alpha : \alpha < \kappa^+ \rangle$ that is linearly ordered and dominating in the partial order of $\prod_{i<\mathrm{cf}(\kappa)} \kappa_i$ by $f \leq g$ iff $|\{ i : f(i) >g(i) \}| < \mathrm{cf}(\kappa)$.  Shelah proved that scales exist for every singular cardinal.
A point $\alpha < \kappa^+$ is called good for a scale on $\kappa$ if there is an unbounded $A \subseteq \alpha$ and an $i < \mathrm{cf}(\kappa)$ such that for all $j > i$, $\langle f_\beta(j) : \beta \in A \rangle$ is a strictly increasing  sequence of ordinals.
The following seems to be folklore:
Theorem (ZFC):  For every scale $\vec F$ for $\aleph_\omega$, there is a club $C \subseteq \aleph_{\omega+1}$ such that every point in $C$ of cofinality at least $\aleph_4$ is good.
Question 1: Is there a proof of this in the literature?  Or can one neatly construct a proof from some lemmas stated in the literature?
Question 2: How does this generalize to larger singular cardinals?
 A: Jing is correct in stating that the result follows from the referenced results in the Abraham-Magidor handbook chapter.
A general theorem, which can be proven in the same way, is the following result:
Theorem: Suppose that $\kappa$ is a singular cardinal and $\vec{f} = \langle f_\alpha \mid \alpha < \lambda \rangle$ is a scale on $\kappa$ (we could have $\lambda > \kappa^+$ here). Then there is a club $C \subseteq \lambda$ such that, for every regular cardinal $\mu$ with $\mathrm{cf}(\kappa) < \mu < \kappa$, and for every $\beta \in C \cap \mathrm{cof}(\mu^{+3})$, $\beta$ is good for $\vec{f}$.
EDIT Actually, upon further reflection, the following slightly more general result, which can again be proven in the same way, is also true:
Theorem: Suppose that $\kappa$ is a singular cardinal and $\vec{f} = \langle f_\alpha \mid \alpha < \lambda \rangle$ is a scale on $\kappa$. Then there is a club $C \subseteq \lambda$ such that, for every regular cardinal $\mu$ with $\mathrm{cf}(\kappa) < \mu < \kappa$, every ordinal $\eta$ with $2 \leq \eta < \mathrm{cf}(\kappa)$, and every $\beta \in C \cap \mathrm{cof}(\mu^{+\eta+1})$, $\beta$ is good for $\vec{f}$.
A: Shelah has also considered this question in his paper [Sh:1008]. The published version indicates that he investigated this from scratch rather than starting with the Sharon-Viale observation on the Abraham-Magidor Handbook article.  I haven't read the paper in detail, so I don't know how much he is able to prove, but I do know he obtains results on the extent $I[\lambda]$, as well as some results on the extent of good points for scales.
[Sh:1008] Shelah, S., Non-reflection of the bad set for $\check{I}_\theta[\lambda]$ and pcf, Acta Math. Hung. 141, No. 1-2, 11-35 (2013). ZBL1324.03014.
