# Can all $\aleph_2$-dense subsets of $\mathbb{R}$ be isomorphic?

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.

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.

• 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 Commented May 29, 2013 at 4:00
• I wonder why that PDF file is called "666". Commented May 29, 2013 at 6:16
• [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. Commented May 29, 2013 at 6:33
• 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. Commented May 29, 2013 at 10:07
• 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.) Commented May 30, 2013 at 4:29

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 $BA_{\aleph_2}.$

• 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! Commented Feb 1, 2015 at 19:53
• Thanks for the references, nevertheless, I have notice that Neeman hasn't publish any paper with the proof. Does anyone knows what happened? Commented Jul 17, 2015 at 20:47
• Hi Iván. I think Neeman's paper is still forthcoming. Commented Jul 18, 2015 at 0:35
• 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. Commented Jul 8, 2017 at 5:29
• Any update regarding Neeman's paper? Commented Sep 29, 2023 at 8:04