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Fresh subsets of ultrapowers


Shelah and Stanley (Proc Am Math Soc 104(3):887–897, 1988) constructed a \(\kappa ^+\)-Aronszjan tree with an ascent path using \(\square _{\kappa }\). We show that \(\square _{\kappa ,2}\) does not imply the existence of Aronszajn trees with ascent paths. The proof goes through an intermediate combinatorial principle, which we investigate further.

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Correspondence to Assaf Shani.

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Shani, A. Fresh subsets of ultrapowers. Arch. Math. Logic 55, 835–845 (2016).

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  • Aronszajn trees
  • Square principles
  • Forcing
  • Fresh subsets

Mathematics Subject Classification

  • 03E05
  • 03E35