Super Black Boxes Revisited

TL;DR

This paper revisits the concept of Super Black Boxes in set theory, focusing on cases where the coloring functions are continuous, and proves their existence for various regular cardinals, with implications for Abelian groups.

Contribution

It extends the theory of Super Black Boxes to continuous colorings and establishes their existence for all regular cardinals, including the critical cases of leph_0 and leph_1.

Findings

Super Black Boxes exist for all regular leph_ and leph_ under continuous coloring assumptions.

The paper proves the existence of multiple ar C-s and free subsets of f^.

Results have implications for the structure of Abelian groups.

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 ${}^\kappa\mu$.