# A club-guessing exercise

I came across this club-guessing exercise on Cardinal Arithmetic by Abraham and Magidor in the Handbook of Set Theory.

Let $\kappa, \lambda$ be regular cardinals $\kappa^{++}<\lambda$ and let $F$ be a (partial) function $:\ \subseteq [\lambda]^{<\kappa}\to \lambda$. For each $\delta\in S^\lambda_{\kappa^{++}}$ (cardinals in $\lambda$ with cofinality $\kappa^{++}$) there exists a club $E_{\delta}$ such that $[E_{\delta}]^{<\kappa}\subseteq dom(F)$. Show $S=\{\alpha\in S^\lambda_\kappa: \exists club \ D\subseteq \alpha \forall a,b\in D (a<b\rightarrow F(\{d\in D: d\leq a\})<b)\}$ is stationary.

What I can do so far is assume that $F$ is total, and then sparse out the elements in the club by the function $F$. This is what I have tried so far: Assume $F$ is total. Fix a club $E\subset \lambda$. Fix a club-guessing sequence $\langle C_i: i\in S^{\kappa^{++}}_\kappa\rangle$. By induction on $\alpha<\kappa^{++}$ build a club set. Suppose we have already defined $F_\alpha\subset E$, if $\sup(F_\alpha)$ is not yet in $F_\alpha$, put it in and go to the next stage (in particular in the limit stage, this make sure the sequence of elements is continuous). Otherwise, list $F_\alpha$ as $\langle f_j: j<\alpha\rangle$. Pick $f_{\alpha+1}$ to be the least in $E$ such that it is greater than $\sup_{k<i}F(\{f_j: j\in C_i \wedge j<\min\{k,\alpha\} \})$ for all $i\in S^{\kappa^{++}}_\kappa$. In this way, we build a club $E_{\delta}\subset \delta\cap E$ of order type $\kappa^{++}$ for some $\delta$ of cofinality $\kappa^{++}$. List them as $\langle e_i: i\in \kappa^{++}\rangle$. Then by club-guessing, there exists some $i\in S^{\kappa^{++}}_\kappa$ such that $C_i\subseteq \{j\in \kappa^{++}: e_j\in E_{\delta}\}$. Then $e_i\in E\cap S$ as desired.

Now to incorporate $E_\delta$, I'm afraid it might mess up with the original indices if I do it as above. Any thoughts would be appreciated!

• Show $S=\ldots$ is not a complete sentence (specifically since you didn't define $S$ before hand). – Asaf Karagila Feb 12 '15 at 17:02
• Oops. My problem of typing. – Jing Zhang Feb 12 '15 at 17:12
• Why are there three votes to close this question? It’s clear, and the OP has certainly put some effort into it. – Brian M. Scott Feb 12 '15 at 21:39
• For the curious, this is Exercise $2.20$ in the cited article, which is available here [DVI]. – Brian M. Scott Feb 12 '15 at 21:52
• @BrianM.Scott I was hoping someone would enlighten me on this as well... – Jing Zhang Feb 12 '15 at 23:55

The modification to the previous argument is when defining continuous $f_{\alpha+1}\in E$ we pick it such that it is greater than $\sup_{k<i} F({f_j:j\in C_i∧j\leq\min\{k,\alpha\}})$ for all $i\in S^{\kappa^{++}}_\kappa$ whenever the value F is defined (if it is not defined we just take whatever available which has not been picked before). Now again we have $\delta$ with cofinality $\kappa^{++}$ such that $F_{\delta}\subset E$ a club. Let $E_\delta\subset \delta$ a club given as in the assumption such that $[E_\delta]^{<\kappa}\subset dom(F)$. Let $D=E_\delta\cap F_{\delta}$ then $D$ is also a club in $\delta$. Also notice that $L=\{j<\kappa^{++}: f_j\in E_{\delta}\}$ then it is a club in $\kappa^{++}$. By club-guessing, for some $i\in S^{\kappa^{++}}_\kappa$we have a club $C_i\subset L$ (so $i \in L$). Verify the property for $P=\{f_k: k\in C_i\}$ club in $f_i$. Given $p<q\in C_i$, by construction we have $F(\{f_d\in F_\delta: d\in C_i \wedge d\leq p\}) < f_{p+1}\leq f_q$. Therefore $f_i\in E\cap S$ and $cf(f_i)=\kappa$.