TY - JOUR
T1 - On two problems of Erdös and Hechler
T2 - New methods in singular madness
AU - Kojman, Menachem
AU - Kubiś, Wiesław
AU - Shelah, Saharon
PY - 2004/11
Y1 - 2004/11
N2 - For an infinite cardinal μ, MAD(μ) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over μ. Erd̈os and Hechler proved in 1973 the consistency of μ, ∈ MAD(μ) for a singular cardinal μ and asked if it was ever possible for a singular μ that μ ∉ MAD(μ), and also whether 2 cf μ < μ ↔ μ ∈ MAD(μ) for every singular cardinal μ. We introduce a new method for controlling MAD(μ) for a singular μ and, among other new results about the structure of MAD(μ) for singular μ, settle both problems affirmatively.
AB - For an infinite cardinal μ, MAD(μ) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over μ. Erd̈os and Hechler proved in 1973 the consistency of μ, ∈ MAD(μ) for a singular cardinal μ and asked if it was ever possible for a singular μ that μ ∉ MAD(μ), and also whether 2 cf μ < μ ↔ μ ∈ MAD(μ) for every singular cardinal μ. We introduce a new method for controlling MAD(μ) for a singular μ and, among other new results about the structure of MAD(μ) for singular μ, settle both problems affirmatively.
KW - Almost disjoint family
KW - Bounding number
KW - Singular cardinal
KW - Smooth pcf scales
UR - http://www.scopus.com/inward/record.url?scp=7444255562&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-04-07580-X
DO - 10.1090/S0002-9939-04-07580-X
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:7444255562
SN - 0002-9939
VL - 132
SP - 3357
EP - 3365
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 11
ER -