The failure of diamond on a reflecting stationary set

Moti Gitik*, Assaf Rinot

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


1. It is shown that the failure of {lozenge, open}S, for a set S ⊆ א ω+1 that reflects stationarily often, is consistent with GCH and APא ω, relative to the existence of a supercompact cardinal. By a theorem of Shelah, GCH and □* λ entails {lozenge, open}S for any S ⊆ λ+ that reflects stationarily often. 2. We establish the consistency of existence of a stationary subset of [א ω+1] ω that cannot be thinned out to a stationary set on which the supfunction is injective. This answers a question of König, Larson and Yoshinobu in the negative. 3. We prove that the failure of a diamond-like principle introduced by Džamonja and Shelah is equivalent to the failure of Shelah's strong hypothesis.

Original languageEnglish
Pages (from-to)1771-1795
Number of pages25
JournalTransactions of the American Mathematical Society
Issue number4
StatePublished - 2012


  • Approachability
  • Diamond
  • Reflection
  • Sap
  • Square
  • Very good scale


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