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Let $S=\{\delta<\omega_2:\text{cf}(\delta)=\omega\}$. A well-known theorem of Shelah tells us that we can find $\langle C_\delta:\delta\in S\rangle$ such that for every club $C\subseteq\omega_2$ there are stationarily many $\delta\in S$ with $C_\delta\subseteq C$, that is, $\langle C_\delta:\delta\in S\rangle$ is a type of club-guessing sequence.

Given $\sigma<\omega_1$, can we in ZFC find a club-guessing sequence as above so that the order-type of each $C_\delta$ is at least $\sigma$? Could we even demand that $C_\delta$ is of order-type $\delta$?

Note that we get such sequences $\langle C_\delta:\delta\in S\rangle$ if $\diamondsuit_S$ holds.

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YES for any given $\sigma<\omega_1$. And you can even have the clubs (somewhat) cohere. I worked out the details in here: http://blog.assafrinot.com/?p=2133

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  • $\begingroup$ Nice! This is exactly what I needed. $\endgroup$ Nov 13 '15 at 21:43

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