Let $\lambda$ be an infinite cardinal. Recall that Weak diamond $\Phi\_S$ on $S\subseteq\lambda^+$ is the following principle:


For every function $F:2^{<\lambda^+}\rightarrow 2$, there exists $g\in 2^{\lambda^+}$ such that for all functions $f:\lambda^+\rightarrow 2$, the set $\{\alpha\in S:~F(f\restriction\alpha)=g(\alpha)\}$ is stationary.


The following theorem is well known:



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> > Theorem: (Devlin-Shelah) For every infinite $\lambda$, $\Phi\_{\lambda^+}\Leftrightarrow 2^\lambda<2^{\lambda^+}$.
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Unlike the above situation, failure of the weak diamond at a stationary set is not enough to get $2^\lambda=2^{\lambda^+}$. In fact, we have the following.



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> > Theorem: (Shelah) The theory ${\rm ZFC}+\neg\Phi\_{S^{\omega\_2}\_\omega}+2^{\omega\_1}<2^{\omega\_2}$ is consistent.
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where $S^{\omega\_2}\_{\omega}=\{\alpha<\omega\_2: {\rm cof}(\alpha)=\omega\}$, and similarly for $S^{\omega\_2}\_{\omega\_1}=\{\alpha<\omega\_2:{\rm cof}(\alpha)=\omega\_1\}$ we have:



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> Theorem:(Shelah) The theory
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Now in the light of above theorems, I would like to know what is the answer to the following question.


### Question: Does simultaneous failure of $\Phi\_{S^{\omega\_2}\_{\omega
}}$ and $\Phi\_{S^{\omega\_2}\_{\omega\_1}}$ imply $2^{\aleph\_1}=2^{\aleph\_2}$?


Edit:
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For a proof of 2nd theorem look at Shelah's paper "More on the weak diamond"-1985. A reference for the 3rd one was given in the comments by Golshani.



Simultaneous failure of weak diamond
*Yes*, it does. 


The collection $I^{\omega\_2}\_{WD}= \{ S \subset \omega\_2: \neg \Phi^{\omega\_2}\_S \}$ of subsets of $\omega\_2$ is an ideal (a proof this fact is provided in the proposition below; moreover I think this result is originally due to Shelah, however a reference escapes me at the moment.) 


Moreover, $\Phi\_{\omega\_2}$ fails, precisely when $I^{\omega\_2}\_{WD}$ is not a proper ideal, which in this case is equivalent to some club subset of $\omega\_2$ being a member of $I^{\omega\_2}\_{WD}$. 


So if $\Phi^{\omega\_2}\_S$ fails for both $E^{\omega\_2}\_\omega$ and $E^{\omega\_2}\_{\omega\_1}$. Then $E^{\omega\_2}\_{\omega}\cup E^{\omega\_2}\_{\omega\_1} =\lim(\omega\_2) \in I^{\omega\_2}\_{WD}$ and this implies the failure of $\Phi\_{\omega\_2}$, which is equivalent to $2^{\aleph\_1} = 2^{\aleph\_2}$.




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**Proposition:** For any cardinal $\lambda$, the set $I^{\lambda^{+}}\_{WD} = \{S \subset \lambda^{+}: \neg \Phi^{\lambda^{+}}\_S \}$ is a $\lambda^{+}$-complete ideal extending the non-stationary ideal on $\lambda^{+}$.


**Proof:** That $I\_{WD}^{\lambda^{+}}$ contains the non-stationary ideal on $\lambda^{+}$ follows by definition. As such, all that remains is to show $I^{\lambda^{+}}\_{WD}$ is an appropriately complete ideal. 


To this end suppose $S \in I^{\lambda^{+}}\_{WD}$ as witnessed by the sequence $\langle F\_\alpha: \alpha \in S \rangle$ where for each $\alpha \in S$, $F\_{\alpha}:\,^{\alpha}2\rightarrow 2$. Then, for every $g:\lambda^{+} \rightarrow 2$, there is some $f:\lambda^{+} \rightarrow 2$ and club $C\_g \subset \lambda^{+}$ such that


$$ \{ \alpha \in S \cap C\_g: F\_\alpha(f\vert\_\alpha) = g(\alpha) \} = \emptyset $$


as such, for every $S\_0 \subset S$ we have $ \{ \alpha \in S\_0 \cap C\_g: F\_\alpha(f\vert\_\alpha) = g(\alpha) \} = \emptyset $, hence the sub-sequence $\langle F\_\alpha : \alpha \in S\_0\rangle$ witnesses $S\_0 \in I^{\lambda^{+}}\_{WD}$. 


Next, fix a bijection $\varphi:\lambda^{+}\times \lambda^{+}\rightarrow \lambda^{+}$ and let $C = \{ \alpha \in \lim(\lambda^{+}): \varphi[\alpha \times \alpha] = \alpha \}$ be the club subset of $\lambda^{+}$ consisting of closure points of $\varphi$.
Now, assume $\{ S\_\gamma: \gamma \in \delta \} \subset I\_{WD}^{\lambda^{+}}$ (with $\delta \in \lambda^{+}$) are stationary, pairwise disjoint, and that for each $\gamma\in \delta$, the sequence $\langle F^{\gamma}\_\alpha : \alpha \in S\_\gamma \rangle$ witnesses the failure of $\Phi^{\lambda^{+}}\_{S\_\gamma}$. 


Letting $E = \bigcup \{ S\_\gamma : \gamma \in \delta \}$, define the sequence $\langle F^{\ast}\_\alpha : \alpha \in E\rangle$ of functions $F^{\ast}\_\alpha : \,^{\alpha}2\rightarrow 2$ as follows, given $\alpha \in E$ and $h \in \,^{\alpha}2$: 


* if $\alpha \in C\cap S\_\gamma$, set $F^{\ast}\_{\alpha}(h) = F^{\gamma}\_{\alpha}([h]\_\gamma)$, where $[h]\_\gamma$ denotes the function defined by $[h]\_\gamma(\xi) = h(\varphi(\gamma,\xi))$, and if $\alpha \not\in C$, simply take $F^{\ast}\_{\alpha}(h) = 0$.


Now, for every $g:\lambda^{+} \rightarrow 2$, there are functions $\{ h\_\gamma : \gamma \in \delta \} \subset \,^{\lambda^{+}}2$ and clubs $\{ C\_\gamma: \gamma \in \delta \} \subset \mathcal{P}(\lambda^{+})$ such that, for each $\gamma \in \delta$,


$$ \{ \alpha \in S\_\gamma \cap C\_\gamma : F^{\gamma}\_{\alpha}(h\_\gamma\vert\_\alpha) = g(\alpha) \} = \emptyset. $$


So, letting $D=\cap \{ C\_\gamma : \gamma \in \delta \}$ and defining $h:\lambda^{+}\rightarrow 2$ by $h(\varphi(\gamma,\xi))=h\_\gamma(\xi)$ for $\gamma \in \delta$, and $h(\varphi(\gamma, \xi)) = 0$ otherwise; we have $\alpha \in S\_\gamma \cap C \cap D\implies $


$$ F^{\ast}\_{\alpha}(h\vert\_\alpha) = F^{\gamma}\_{\alpha}([h]\_{\gamma}\vert\_\alpha)=F^{\gamma}\_{\alpha}(h\_{\gamma}\vert\_\alpha)=1 - g(\alpha).$$
Therefore, $ \{ \alpha \in E\cap C \cap D: F^{\ast}\_{\alpha}(h\vert\_\alpha) = g(\alpha) \} = \emptyset $, and it immediately follows that $$E=\cup \{ S\_\gamma: \gamma \in \delta \} \in I^{\lambda^{+}}\_{WD}$$ $\square$.


