I've been told once or twice that the following holds:
There is a model of $ZFC+MA+\neg CH$ in which there is a $\mathfrak{c}$-universal linear order embedded in $(\omega^\omega, \le^\ast)$
Moreover, several people have attributed the construction of such a model to Hugh Woodin. The context that seems the most natural for this construction to appear is in his work on automatic-continuity, however for the life of me, I can't seem to find it.
It might also be the case that the embedding is into $\mathcal{P}(\omega)/fin$.
Does anybody know of a reference for this? Or if Woodin did indeed construct it (I only ask because my searches have yielded little fruit)?
Edit:
In this context a $\mathfrak{c}$-universal linear order $(L, \prec)$ has the following property: $\vert L \vert \le \mathfrak{c}$ and for every linear order $(\ell, <) $, with $\vert \ell \vert \le \mathfrak{c}$ there is an embedding $\varphi_\ell:\ell \rightarrow L$ respecting the linear order of $\ell$.
Thought I should share what I can find/know:
So far I've been able to find models for the following
$ZFC+ \neg CH$ and there is such a $\mathfrak{c}$-universal linear order in $(\omega^\omega,\le^\ast)$.
This is due to Laver, http://www.sciencedirect.com/science/article/pii/S0049237X08716306
In addition, there is also
$ZFC+MA+\neg CH$ and no such order is embedded in $(\omega^\omega,\le^\ast)$
There seem to be several models for this one, Laver cites two, one from Solovay and one from Kunen. I can only assume the Solovay model is the same constructed in Theorem 5.7 (p. 201, "Hausdroff Gaps and Limits", by Frankiewicz and Zbierski)
In addition there is
$CH\implies $ $(\omega^\omega,\le^\ast)$ is contains a $\mathfrak{c}$-universal linear order
Which should be contained in (or atleast hinted at in) "Model Theory", by Chang and Keisler. If not there, the lack of $(\omega,\omega)$-gaps in $(\omega^\omega,\le^\ast)$ should produce it rather quickly.
So yeah, it would seem that the only link missing in this puzzle is the one I can't find the reference for.
