Super Black Boxes Revisited

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} \emph{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 ${}^\kappa\mu$.