The following is a theorem of Baumgartner, Hajnal, and Mate:


Suppose $J$ is a normal ideal on $\omega\_1$ which is nowhere $\omega\_1$-dense. Then for any sequence $\langle A\_\alpha : \alpha < \omega\_1 \rangle$ of $J$-positive sets, there is a sequence $\langle B\_\alpha : \alpha < \omega\_1 \rangle$ of pairwise disjoint positive sets such that $B\_\alpha \subseteq A\_\alpha$ for all $\alpha$.


The proof is specific to $\omega\_1$. But I conjecture that ZFC+CH proves the same statement with $\omega\_1$ replaced by $\omega\_2$ throughout.


Is the conjecture true? Are there any related results you think might shed light on this?



Disjoint refinements in $P(\kappa)/J$
In my [thesis](http://www.math.uci.edu/~meskew/thesis.pdf) I proved that my conjecture is false. It's not so simple-- the proof ties together chapter 2 and sections 5.1, 5.2, and 6.3.


