If we assume MA+¬CH, then every boolean algebra with cardinality smaller than the continuum embeds in ℘(ω)/Fin. A proof of this result can be found in Theorem 1.1, Chapter 8 of the book "Hausdorff gaps and limits". In the paper "R. Frankiewicz, Some remarks on embeddings of boolean algebras and topological spaces II, Fund. Math, 126 (1), 1985, 63-68.", the author showed that it is consistent with MA+¬CH the existence of a boolean algebra with cardinality equal to the continuum which does not embed in ℘(ω)/Fin. 


Is it consistent with $MA+\neg CH$ that every boolean algebra with cardinaly equal to the continuum embeds in ℘(ω)/Fin?



Embeddings of Boolean algebras in $\wp(\omega)/Fin$
In the paper [Embedding of Boolean algebras in $Ρ(ω)/$fin](https://eudml.org/doc/211828) the following partial result is proved:


**Theorem** There is a model of $ZFC$ with arbitrarily large continuum in which each Boolean algebra $B$ of cardinality $\leq 2^{\aleph\_0}$ can be embedded into $P(ω)/$fin. In addition, Martin's axiom for $σ$-linked orderings holds in the model.


You may also look at [On automorphisms of Boolean algebras embedded in P (ω)/fin](http://matwbn.icm.edu.pl/ksiazki/fm/fm150/fm15023.pdf), where a model as above is constructed in which the following additional property holds: every automorphism of $B$ extends to an automorphism of $P(ω)/$fin.


