Super black box (formerly: Middle diamond)

Abstract

This is a slightly corrected version of an old work. Under certain cardinal arithmetic assumptions, we prove that for every large enough regular $λ$ cardinal, for many regular $κ< λ$, many stationary subsets of $λ$ concentrating on cofinality $κ$ have super BB. In particular, we have the super BB on $\{δ< λ\colon cf(δ) = κ\}$. 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&#34; which is now replaced by ``super BB&#39;&#39;, that is, ``super black box&#39;&#39;, in order to be consistent with other papers (see [Sh:898]).