TY - JOUR
T1 - On the consistency strength of the Milner-Sauer conjecture
AU - Rinot, Assaf
PY - 2006/7
Y1 - 2006/7
N2 - In their paper from 1981, Milner and Sauer conjectured that for any poset ( P, ≤) , if cf ( P, ≤ ) = λ > cf (λ ) = K , then P must contain an antichain of cardinality K . The conjecture is consistent and known to follow from GCH-type assumptions. We prove that the conjecture has large cardinals consistency strength in the sense that its negation implies, for example, the existence of a measurable cardinal in an inner model. We also prove that the conjecture follows from Martin's Maximum and holds for all singular λ above the first strongly compact cardinal.
AB - In their paper from 1981, Milner and Sauer conjectured that for any poset ( P, ≤) , if cf ( P, ≤ ) = λ > cf (λ ) = K , then P must contain an antichain of cardinality K . The conjecture is consistent and known to follow from GCH-type assumptions. We prove that the conjecture has large cardinals consistency strength in the sense that its negation implies, for example, the existence of a measurable cardinal in an inner model. We also prove that the conjecture follows from Martin's Maximum and holds for all singular λ above the first strongly compact cardinal.
KW - Large cardinals
KW - Poset
KW - Singular cofinality
KW - Singular density
UR - http://www.scopus.com/inward/record.url?scp=33748421414&partnerID=8YFLogxK
U2 - 10.1016/j.apal.2005.09.012
DO - 10.1016/j.apal.2005.09.012
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AN - SCOPUS:33748421414
SN - 0168-0072
VL - 140
SP - 110
EP - 119
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 1-3
ER -