Super black box (formerly: Middle diamond)

This is a slightly corrected version of an old work. Under certain cardinal arithmetic assumptions, we prove that for every large enough regular $\lambda$ cardinal, for many regular $\kappa < \lambda$, many stationary subsets of $\lambda$ concentrating on cofinality $\kappa$ have super BB. In particular, we have the super BB on $\{\delta < \lambda \colon cf(\delta) = \kappa\}$. This is a strong negation of uniformization. We have added some details. Works continuing it are [Sh:898] and [Sh:1028]. We thank Ari Brodski and Adi Jarden for their helpful comments. In this paper we had earlier used the notion ``middle diamond" which is now replaced by ``super BB'', that is, ``super black box'', in order to be consistent with other papers (see [Sh:898]).