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 $\overline{C}=⟨C_{\delta }:\delta \in S⟩$, where $C_{\delta }$ is a club of $\delta$ of order type $\kappa$, for every coloring $\overline{F}=⟨F_{\delta }:\delta \in S⟩$ with $F_{\delta }:{}^{C_{\delta }}\lambda \to \theta$, there exists $⟨c_{\delta }:\delta \in S⟩\in {}^S\theta$ 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 $$⟨F_{\delta ,\beta }^{\prime }(f\upharpoonright (C_{\delta }\cap \beta )):\beta \in C_{\delta }⟩.$$ 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 ${}^{\kappa }\mu$.