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.



Club-guessing at $\omega_2$
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>


