proof of $\mathrm{\u25c7}$ is equivalent to $\mathrm{\u2663}$ and continuum hypothesis
The proof that ${\mathrm{\u25c7}}_{S}$ implies both ${\mathrm{\u2663}}_{S}$ and that for every $$, ${2}^{\lambda}\le \kappa $ are given in the entries for ${\mathrm{\u25c7}}_{S}$ and ${\mathrm{\u2663}}_{S}$.
Let $A={\u27e8{A}_{\alpha}\u27e9}_{\alpha \in S}$ be a sequence which satisfies ${\mathrm{\u2663}}_{S}$.
Since there are only $\kappa $ bounded^{} subsets of $\kappa $, there is a surjective function $f:\kappa \to \mathrm{Bounded}(\kappa )\times \kappa $ where $\mathrm{Bounded}(\kappa )$ is the bounded subsets of $\kappa $. Define a sequence $$ by ${B}_{\alpha}=f(\alpha )$ if $$ and $\mathrm{\varnothing}$ otherwise. Since the set of $({B}_{\alpha},\lambda )\in \mathrm{Bounded}(\kappa )\times \kappa $ such that ${B}_{\alpha}=T$ is unbounded^{} for any bounded subset $T$, it follow that every bounded subset of $\kappa $ occurs $\kappa $ times in $B$.
We can define a new sequence, $D={\u27e8{D}_{\alpha}\u27e9}_{\alpha \in S}$ such that $x\in {D}_{\alpha}\leftrightarrow x\in {B}_{\beta}$ for some $\beta \in {A}_{\alpha}$. We can show that $D$ satisfies ${\mathrm{\u25c7}}_{S}$.
First, for any $\alpha $, $x\in {D}_{\alpha}$ means that $x\in {B}_{\beta}$ for some $\beta \in {A}_{\alpha}$, and since ${B}_{\beta}\subseteq \beta \in {A}_{\alpha}\subseteq \alpha $, we have ${D}_{\alpha}\subseteq \alpha $.
Next take any $T\subseteq \kappa $. We consider two cases:
$T$ is bounded
The set of $\alpha $ such that $T={B}_{\alpha}$ forms an unbounded sequence ${T}^{\prime}$, so there is a stationary ${S}^{\prime}\subseteq S$ such that $\alpha \in {S}^{\prime}\leftrightarrow {A}_{\alpha}\subset {T}^{\prime}$. For each such $\alpha $, $x\in {D}_{\alpha}\leftrightarrow x\in {B}_{i}$ for some $i\in {A}_{\alpha}\subset {T}^{\prime}$. But each such ${B}_{i}$ is equal to $T$, so ${D}_{\alpha}=T$.
$T$ is unbounded
We define a function $j:\kappa \to \kappa $ as follows:

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$j(0)=0$

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To find $j(\alpha )$, take $$. This is a bounded subset of $\kappa $, so is equal to an unbounded series of elements of $B$. Take $j(\alpha )=\gamma $, where $\gamma $ is the least number greater than any element of $$ such that $$.
Let ${T}^{\prime}=\mathrm{range}(j)$. This is obviously unbounded, and so there is a stationary ${S}^{\prime}\subseteq S$ such that $\alpha \in {S}^{\prime}\leftrightarrow {A}_{\alpha}\subseteq {T}^{\prime}$.
Next, consider $C$, the set of ordinals^{} less than $\kappa $ closed under $j$. Clearly it is unbounded, since if $$ then $j(\lambda )$ includes $j(\alpha )$ for $$, and so induction^{} gives an ordinal greater than $\lambda $ closed under $j$ (essentially the result of applying $j$ an infinite^{} number of times). Also, $C$ is closed: take any $c\subseteq C$ and suppose $\mathrm{sup}(c\cap \alpha )=\alpha $. Then for any $$, there is some $\gamma \in c$ such that $$ and therefore $$. So $\alpha $ is closed under $j$, and therefore contained in $C$.
Since $C$ is a club, ${C}^{\prime}=C\cap {S}^{\prime}$ is stationary. Suppose $\alpha \in {C}^{\prime}$. Then $x\in {D}_{\alpha}\leftrightarrow x\in {B}_{\beta}$ where $\beta \in {A}_{\alpha}$. Since $\alpha \in {S}^{\prime}$, $\beta \in \mathrm{range}(j)$, and therefore ${B}_{\beta}\subseteq T$. Next take any $x\in T\cap \alpha $. Since $\alpha \in C$, it is closed under $j$, hence there is some $\gamma \in \alpha $ such that $j(x)\in \gamma $. Since $\mathrm{sup}({A}_{\alpha})=\alpha $, there is some $\eta \in {A}_{\alpha}$ such that $$, so $j(x)\in \eta $. Since $\eta \in {A}_{\alpha}$, ${B}_{\eta}\subseteq {D}_{\alpha}$, and since $\eta \in \mathrm{range}(j)$, $j(\delta )\in {B}_{\eta}$ for any $$, and in particular $x\in {B}_{\eta}$. Since we showed above that ${D}_{\alpha}\subseteq \alpha $, we have ${D}_{\alpha}=T\cap \alpha $ for any $\alpha \in {C}^{\prime}$.
Title  proof of $\mathrm{\u25c7}$ is equivalent^{} to $\mathrm{\u2663}$ and continuum hypothesis^{} 

Canonical name  ProofOfDiamondIsEquivalentToclubsuitAndContinuumHypothesis 
Date of creation  20130322 12:53:57 
Last modified on  20130322 12:53:57 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  6 
Author  Henry (455) 
Entry type  Proof 
Classification  msc 03E45 
Synonym  proof that diamond is equivalent to club and continuum hypothesis 
Related topic  Diamond^{} 
Related topic  Clubsuit 