Sh:1268

• Shelah, S. Super black boxes revisited. Preprint. arXiv: 2602.09592
• Abstract:
  Let \kappa , \theta < \lambda be cardinals, with \lambda and \kappa regular. Concentrating on a simple case, 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 (which have not been covered yet).

  Here we restrict ourselves to the case where \overline F is a so-called continuous coloring, which includes the case where F_\delta is computed from some \big\langle F_{\delta,\beta}'(f \upharpoonright (C_\delta \cap \beta)) : \beta \in C_\delta \big\rangle.

  This covers the cases we have in mind. We mainly prove results without any other caveats: e.g.

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

  We also deal with having multiple \bar C-s, and the existence of quite free subsets of {}^\kappa\mu.

• Version 2026-01-18 (39p)

@article{Sh:1268,
 author = {Shelah, Saharon},
 title = {{Super black boxes revisited}},
 note = {\href{https://arxiv.org/abs/2602.09592}{arXiv: 2602.09592}},
 arxiv_number = {2602.09592}
}