### Martin's axiom and separated MAD families

by Dow and Shelah. [DwSh:985]

Rend. Circ. Mat. Palermo, 2012

Two families A, B of subsets of omega
are said to be separated if there is a subset of omega which mod
finite contains every member of A and is almost disjoint
from every member of B . If A and B
are countable disjoint subsets of an almost disjoint family, then
they are separated. Luzin gaps are well-known examples of
of omega_1-sized subfamilies of an almost disjoint family which can
not be separated. An almost disjoint family will be said to be
omega_1-separated if any disjoint pair of {<=} omega_1-sized
subsets are separated. It is known that the proper forcing axiom
(PFA) implies that no maximal almost disjoint family is
{<=} omega_1-separated. We prove that this does not follow
from Martin's Axiom.

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