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](http://matwbn.icm.edu.pl/ksiazki/fm/fm79/fm79111.pdf), 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.



Can all $\aleph_2$-dense subsets of $\mathbb{R}$ be isomorphic?
The consistency of *All $\aleph\_2$-dense sets are order-isomorphic* is problem 15.14 in [Arnold Miller's list of problems](http://www.math.wisc.edu/~miller/res/problems.pdf) (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.


