TY - JOUR

T1 - Antichains in partially ordered sets of singular cofinality

AU - Rinot, Assaf

PY - 2007/7

Y1 - 2007/7

N2 - In their paper from 1981, Milner and Sauer conjectured that for any poset 〈P,≤〉, if cf〈P,≤〉=λ>cfλ=κ, then P must contain an antichain of size κ. We prove that for λ > cf(λ)=κ, if there exists a cardinalμ<λ such that cov(λ, μ, κ, 2)=λ, then any poset of cofinality λ contains λ κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.

AB - In their paper from 1981, Milner and Sauer conjectured that for any poset 〈P,≤〉, if cf〈P,≤〉=λ>cfλ=κ, then P must contain an antichain of size κ. We prove that for λ > cf(λ)=κ, if there exists a cardinalμ<λ such that cov(λ, μ, κ, 2)=λ, then any poset of cofinality λ contains λ κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.

KW - Antichain

KW - Poset

KW - Singular cofinality

UR - http://www.scopus.com/inward/record.url?scp=34249078594&partnerID=8YFLogxK

U2 - 10.1007/s00153-007-0049-z

DO - 10.1007/s00153-007-0049-z

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AN - SCOPUS:34249078594

SN - 0933-5846

VL - 46

SP - 457

EP - 464

JO - Archive for Mathematical Logic

JF - Archive for Mathematical Logic

IS - 5-6

ER -