On the consistency strength of the Milner-Sauer conjecture

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Abstract

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.

Original languageEnglish
Pages (from-to)110-119
Number of pages10
JournalAnnals of Pure and Applied Logic
Volume140
Issue number1-3
DOIs
StatePublished - Jul 2006
Externally publishedYes

Keywords

  • Large cardinals
  • Poset
  • Singular cofinality
  • Singular density

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