Super Black Boxes Revisited
Saharon Shelah
TL;DR
This work revisits the Super Black Box framework for cardinals $(\lambda,\kappa,\theta)$, focusing on continuous colorings and stationary sets to develop BB and DBB machinery. It introduces BB-parameters and Sep-conditions to formulate and prove BB^1- and DBB-properties, establishing existence results under minimal cardinal-arithmetic hypotheses (e.g., $\lambda=cf(2^\mu)$) and providing tools to handle multiple $C_\delta$-families. The results yield constructions of quite free subsets $\Lambda \subseteq {}^\kappa\mu$, with strong limit singular $\mu$ giving $\mu^+$-free sets of size $\lambda$, and have implications for abelian group constructions (e.g., trivial dual). Overall, the paper extends Shelah’s Black Box framework beyond earlier bounds, offering inner-model–leaned combinatorial devices that support transfinite constructions under weaker GCH assumptions.
Abstract
Let $ κ, θ< λ$ be cardinals, with $λ$ and $κ$ regular. Concentrating on a simple case, we say that the triple $(λ,κ,θ)$ has a Super Black Box when the following holds. For some stationary $S \subseteq \{δ< λ: cf(δ) = κ\}$ and $\overline C = \langle C_δ: δ\in S \rangle$, where $C_δ$ is a club of $δ$ of order type $κ$, for every coloring $\overline F = \langle F_δ: δ\in S \rangle$ with $F_δ: {}^{C_δ}λ\to θ$, there exists $\langle c_δ: δ\in S\rangle \in {}^S\!θ$ such that for every $f : λ\to θ$, for stationarily many $δ\in S$, we have $F_δ(f \upharpoonright C_δ) = c_δ$. In an earlier work, it was proved (along with much more) that for a class of cardinals $λ$ this holds for many pairs $(κ,θ)$. E.g.~$κ< \aleph_ω$ is large enough, and $\beth_ω(θ) < λ$. However, the most interesting cases (at least with regards to Abelian groups) are $κ= \aleph_0,\aleph_1$ (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 $F_δ$ is computed from some $$ \big\langle F_{δ,β}'(f \upharpoonright (C_δ\cap β)) : β\in C_δ\big\rangle. $$ This covers the cases we have in mind. We mainly prove results without any other caveats: e.g. For every regular $κ$ and $θ$ there exists such a $λ$. We also deal with having multiple {$\bar C$-s}, and the existence of quite free subsets of ${}^κμ$.