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It is quite well known that

Any FS (finite support) iteration of length $<\mathfrak{c}^+$ of $\sigma$-centered posets is $\sigma$-centered (see e.g. here).

Now consider the following question: whenever $\delta<\mathfrak{c}^+$ is a limit ordinal of uncountable cofinality and $\langle\mathbb{P}_{\alpha}:\alpha<\delta\rangle$ is a sequence of $\sigma$-centered posets such that

(i) $\mathbb{P}_\alpha$ is a complete subposet of $\mathbb{P}_\beta$ for all $\alpha<\beta<\delta$, and

(ii) $\mathbb{P}_\gamma=\bigcup_{\alpha<\gamma}\mathbb{P}_\alpha$ for any limit $\gamma<\delta$,

do we have that $\mathbb{P}_\delta:=\bigcup_{\alpha<\delta}\mathbb{P}_\alpha$ is $\sigma$-centered? Is this correct in the case $\delta=\omega_1$?

Note that this may not be a consequence of the result mentioned at the beginning since the quotient of two $\sigma$-centered posets is not necessarily (forced to be) $\sigma$-centered.

Any help with this problem will be highly appreciated.

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  • $\begingroup$ I feel like there is a counter-example arising from something like, the failure of uniformization for a ladder-system on $\omega_1$. However a specific example escapes me at the moment. $\endgroup$
    – Not Mike
    Commented Mar 9, 2018 at 0:53
  • $\begingroup$ Can you give an example of a composition $P*Q$ which is $\sigma$-centered, where $P$ forces that $Q$ is not $\sigma$-centered? $\endgroup$
    – Goldstern
    Commented Mar 18, 2019 at 15:15
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    $\begingroup$ @Goldstern The most natural (but in this case artificial) is, under $\text{MA}_{\aleph_1}$, $C*T$ where $C$ is Cohen forcing and $T$ is the Suslin tree it adds (recall that $\text{MA}_{\aleph_1}$ implies that any poset of size $\aleph_1$ is $\sigma$-centered). Now, assuming $\text{MA}_{\aleph_1}$, perform a finite support iteration of length $\omega_2$ using $C*T$. I wonder whether the resulting poset is $\sigma$-centered $\endgroup$
    – dragoon
    Commented Mar 23, 2019 at 1:28
  • $\begingroup$ Can't you take a strongly proper forcing $P$ of size $\aleph_1$ that is not c.c.c, then consider some increasing internally approachable countable elementary submodels $\langle M_\alpha: \alpha<\omega_1\rangle$, then $\langle P\cap M_\alpha: \alpha<\omega_1\rangle$ seems to work. $\endgroup$
    – Otto
    Commented May 21, 2020 at 8:58

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