Sh:1268

• Shelah, S. Super black boxes revisited. Preprint.
• Abstract:
  

  Let \lambda > \kappa > \theta be cardinals, with \lambda and \kappa regular. We say that the triple (\lambda,\kappa,\theta) has a Super Black Box when the following holds.

  For some stationary S \subseteq \{\delta < \lambda : cf(\delta) = \kappa\} and \overline C = \langle C_\delta : \delta \in S \rangle, where C_\delta is a club of \delta of order type \kappa, for every coloring \overline F = \langle F_\delta : \delta \in S \rangle with F_\delta : {}^{C_\delta}\lambda \to \theta, there exists \langle c_\delta : \delta \in S\rangle \in {}^S\!\theta such that for every f : \lambda \to \theta, for stationarily many \delta \in S, we have F_\delta(f \upharpoonright C_\delta) = c_\delta.

  In an earlier work, it was proved (along with much more) that for a class of cardinals \lambda this holds for many pairs (\kappa,\theta). E.g. \kappa < \aleph_\omega is large enough, and \beth_\omega(\theta) < \lambda. However, the most interesting cases (at least with regards to Abelian groups) are \kappa = \aleph_0,\aleph_1.

  Here we restrict ourselves to \overline F a so-called continuous coloring, which includes the case where F_\delta just codes f \upharpoonright C_\delta for some f \in {}^\lambda\theta. We mainly prove results without any other caveats: e.g.

  • For every regular \kappa and \theta there exists a \lambda.

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@article{Sh:1268,
 author = {Shelah, Saharon},
 title = {{Super black boxes revisited}}
}