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Let $\kappa$ be an infinite cardinal. For a subset $A \subseteq \mathbb{R}$, we say that $A$ is $\kappa$-dense if $|A \cap (a, b)| = \kappa$ for every interval $(a, b)$. By Cantor, any two $\aleph_0$-dense sets are order-isomorphic.

If the CH holds, then there are many non-isomorphic $\aleph_1$-dense sets. However, Baumgartner showed that it is consistent with $2^{\aleph_0}=\aleph_2$ that all $\aleph_1$-dense sets are isomorphic, giving a generalization of Cantor's result. My question is, can this be pushed further? Is it consistent with $2^{\aleph_0} \geq \aleph_3$ that all $\aleph_2$-dense sets of reals are isomorphic? And what about for larger cardinals still? Can we have the result for $\aleph_1$-dense sets and $\aleph_2$-dense sets simultaneously?

Baumgartner himself asked if all $\aleph_2$-dense sets of reals could be isomorphic in his original paper, which I found linked here on MO (all thanks to François Dorais). But I have been unable to find out if his question has subsequently been answered. If anyone knows the answer, or can provide a reference, I would be grateful.

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The consistency of All $\aleph_2$-dense sets are order-isomorphic is problem 15.14 in Arnold Miller's list of problems (which he keeps up-to-date), and there is no indication there that any progress has been made on it. That's at least strong evidence that it's still open.

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    $\begingroup$ Shelah also mentions it among the most important open problems in "On what I do not understand (and have something to say): I"shelah.logic.at/files/666.pdf $\endgroup$
    – Péter Komjáth
    May 29, 2013 at 4:00
  • $\begingroup$ I wonder why that PDF file is called "666". $\endgroup$
    – Andrej Bauer
    May 29, 2013 at 6:16
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    $\begingroup$ [666] is Shelah's 666th paper, but of course there is some wiggle room that he uses to assign particularly interesting numbers to special papers. $\endgroup$
    – Stefan Geschke
    May 29, 2013 at 6:33
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    $\begingroup$ I think the assignment of numbers to Shelah's papers is not done by Shelah himself but by Andrzej Roslanowski, who maintains the archive of Shelah's papers. $\endgroup$
    – Andreas Blass
    May 29, 2013 at 10:07
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    $\begingroup$ Not quite, Andreas! Saharon was quite involved with the choice of "666" for that paper...he thought it a fine joke! (He was working on this paper when I first started working for him in Jerusalem.) $\endgroup$
    – Todd Eisworth
    May 30, 2013 at 4:29
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In section 5 (Concluding remarks and recent developments) of the paper Baumgartner's isomorphism problem for $\aleph_2$-dense suborders of $\mathbb{R}$, by Moore and Todorcevic, the following is stated:

He (Itay Neeman) moreover tentatively announced that, in spite of the gap in the present paper, he was able to show that if there is a weakly compact cardinal, then there is a forcing extension in which $BA_{\aleph_2}$ and $MA_{\aleph_2}$ both hold.

If the claim is true, it means that the problem is solved now!!!

Also in the notes Reflection of clubs, and forcing principles at $\aleph_2$ by Itay Neeman, the last theorem stated the consistency of $BA_{\aleph_2}.$

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  • $\begingroup$ Mohammad: Thank you for posting those slides. It's my understanding that people in the know, including Justin Moore (though I do not speak for him), believe that Neeman has succeeded in solving the problem. I spoke with Itay himself after I heard about his Luminy talk and he tentatively affirmed his solution. We'll have to wait for his paper, but it would seem likely the problem is resolved! $\endgroup$
    – Garrett Ervin
    Feb 1, 2015 at 19:53
  • $\begingroup$ Thanks for the references, nevertheless, I have notice that Neeman hasn't publish any paper with the proof. Does anyone knows what happened? $\endgroup$
    – Iván Ongay Valverde
    Jul 17, 2015 at 20:47
  • $\begingroup$ Hi Iván. I think Neeman's paper is still forthcoming. $\endgroup$
    – Garrett Ervin
    Jul 18, 2015 at 0:35
  • $\begingroup$ At the 6ESTC in Budapest Neeman just gave a lecture on this. He indicated that he is close to finish the paper. The lecture showed some of the many ideas of the long and complicated proof. $\endgroup$
    – Péter Komjáth
    Jul 8, 2017 at 5:29

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