TY - JOUR
T1 - Martin's Axiom and separated mad families
AU - Dow, Alan
AU - Shelah, Saharon
PY - 2012/4
Y1 - 2012/4
N2 - Two families A,B of subsets of ω are said to be separated if there is a subset of ω 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 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.
AB - Two families A,B of subsets of ω are said to be separated if there is a subset of ω 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 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.
KW - Martin's Axiom
KW - Maximal almost disjoint family
UR - https://www.scopus.com/pages/publications/84874500198
U2 - 10.1007/s12215-011-0078-7
DO - 10.1007/s12215-011-0078-7
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AN - SCOPUS:84874500198
SN - 0009-725X
VL - 61
SP - 107
EP - 115
JO - Rendiconti del Circolo Matematico di Palermo
JF - Rendiconti del Circolo Matematico di Palermo
IS - 1
ER -