TITLE: Embeddings of Boolean algebras in $\wp(\omega)/Fin$
QUESTION [9 upvotes]: 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?

REPLY [4 votes]: In the paper Embedding of Boolean algebras in $Ρ(ω)/$fin 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, 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.