We know that, under MA, every linear order $(X,\le)$ with $|X|<\mathfrak c$ embedds in $\wp(\omega)/Fin$. Does this hold for linear orders with cardinality $\mathfrak c$?



Embeddings of linear orders in $\wp(\omega)/Fin$ under Martin's axiom
This is actually independent of $MA+\neg CH$.


For example, under $MA+OCA$ there are no gaps in $\mathcal{P}(\omega)/fin$ of type $(\mathfrak{c}, \mathfrak{c}^\ast)$, which excludes the possibility that every linear order of size $\mathfrak{c}$ embeds. This is because assuming $MA$, the linear order $L=(2^{<\mathfrak{c}}, <\_\text{lex})$ has size $\mathfrak{c}$ and $2^{\mathfrak{c}}>\mathfrak{c}$ gaps of type $(\mathfrak{c}, \mathfrak{c}^\ast)$; hence any embedding of $L$ into $\mathcal{P}(\omega)/fin$ witnesses the existence of a gap of type $(\mathfrak{c}, \mathfrak{c}^\ast)$ in $\mathcal{P}(\omega)/fin$.


That said, the consistency of $MA+$"every linear order of size $\mathfrak{c}$ embeds into $\mathcal{P}(\omega)/fin$" is attributed to H. Woodin 
(see this question "[$\mathfrak{c}$-universal linear order](https://mathoverflow.net/questions/91027/mathfrakc-universal-linear-order)") 


To construct such a model, you use the following two facts,


* Any $c.c.c.$ forcing which adds a generic gap of type $(\omega\_1, \omega\_1^{\ast})$ to a *ground-model* linear order adds an $\omega\_1$-branch to some *ground-model* Souslin-tree.
* The partial order which specializes a Souslin-tree with finite conditions does not add a branch to any ground-model $\omega\_1$-tree.


and construct an iteration which interleaves Souslin-free forcings with those that split gaps in the $\mathfrak{c}$-universal linear order you are building.


