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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