## Abstract

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 language | English |
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Pages (from-to) | 1771-1795 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society |

Volume | 364 |

Issue number | 4 |

DOIs | |

State | Published - 2012 |

## Keywords

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