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Shelah proved (paper 667) that if GCH holds and $\lambda$ is singular, then for every stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder system $\{ c_\alpha : \alpha \in S \}$ and a coloring of the ladders $\{ f_\alpha : \alpha \in S \}$ in 2 colors (where each $f_\alpha : c_\alpha \to 2$), which cannot be uniformized. This means that there is not any $F : \lambda^+ \to 2$ such that for all $\alpha \in S$, there is a set $b_\alpha$ of size $<\text{cf}(\lambda)$ such that $F \restriction (c_\alpha \setminus b_\alpha) = f_\alpha \restriction (c_\alpha \setminus b_\alpha)$. This is in contrast to the case of $\lambda$ regular, where it is always possible to force ladder system uniformization on an appropriately stationary-costationary set and preserve GCH.

Shelah built the ladder roughly by diagonalizing against all small approximations, making sure that for all $F : \lambda^+ \to 2$, there is $\alpha \in S$ such that, if $\langle\beta_i : i < \text{cf}(\lambda) \rangle$ enumerates $c_\alpha$, then $F(\beta_{2i}) = F(\beta_{2i+1})$ for all $i<\text{cf}(\lambda)$. This guarantees that the “even-odd” coloring of the ladders cannot agree mod bounded with a single $F: \lambda^+ \to 2$.

Question: Is it consistent with GCH that there is a singular $\lambda$ and a stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$ such that for all ladder systems $\langle c_\alpha : \alpha \in S \rangle$ and all “constant colorings” $\langle f_\alpha : \alpha \in S \rangle$ (where each $f_\alpha : c_\alpha \to 2$ is constant), there is a uniformization $F : \lambda^+ \to 2$?

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  • $\begingroup$ This question looks familiar! Just a remark: In Shelah's "Successors of Singular Cardinals: Combinatorics ... (etc)" [Sh:667] Application 2.12 (b), there is perhaps a typo,as the remark "we can omit the assumption (iii)" does not make sense, and if he meant (iv) instead of (iii) (which makes sense given the context) then the answer is yes. Of course there is a typo in the portion of the abstract dealing with this matter as well, which states that Section 2 contains a result that directly contradicts the main Theorem of Section 1. $\endgroup$ Jul 9, 2019 at 18:47
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    $\begingroup$ Also, I emailed Shelah already today to get clarification. $\endgroup$ Jul 9, 2019 at 19:00
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    $\begingroup$ Have you looked at the Gitik-Rinot paper where they got $\neg\diamondsuit(S) + GCH$ for some stationary $S$ consisting of ordinals of critical cofinality? Maybe adapting the forcing there to force uniformization instead of just failure of diamond may give a positive answer. $\endgroup$
    – Otto
    Dec 9, 2019 at 21:24
  • $\begingroup$ @Otto Good suggestion, thanks. $\endgroup$ Dec 10, 2019 at 10:00

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