Abstract
Two families \(\mathcal{A}, \mathcal{B}\) of subsets of ω are said to be separated if there is a subset of ω which mod finite contains every member of \(\mathcal{A}\) and is almost disjoint from every member of \(\mathcal{B}\). If \(\mathcal{A}\) and \(\mathcal{B}\) are countable disjoint subsets of an almost disjoint family, then they are separated. Luzin gaps are well-known examples of ω1-sized subfamilies of an almost disjoint family which can not be separated. An almost disjoint family will be said to be ω1-separated if any disjoint pair of ≤ω1-sized subsets are separated. It is known that the proper forcing axiom (PFA) implies that no maximal almost disjoint family is ≤ω1-separated. We prove that this does not follow from Martin’s Axiom.
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