Ketonen's question and other cardinal sins
Assaf Rinot, Zhixing You, Jiachen Yuan
TL;DR
The paper investigates three compactness phenomena at uncountable cardinals by exploiting a unifying intersection-model technique: (i) Ketonen's question about indecomposable ultrafilters on weakly compact cardinals, (ii) the behavior of λ-strongly compact cardinals under forcing, and (iii) the C-sequence number χ(κ). It develops a framework of direct-limit forcing and intersection models to construct limit ultrafilters and ascent-path structures, then applies it to produce a model where a weakly compact κ carries an indecomposable ultrafilter but is not measurable, and to realize singular least λ-strongly-compact cardinals for multiple λ's. It further shows χ(κ) can be made any prescribed δ<κ while preserving cofinalities, via a two-block forcing using indexed square sequences and threading. The results highlight the versatility of intersection models in producing new compactness configurations and provide new proofs for related results (e.g., super Ramsey phenomena).
Abstract
Answering a question of Ketonen from the late 1970's, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing the limit of a decreasing sequence of models of ZFC. The utility of this proof technique is demonstrated further in this paper, where a problem by Bagaria and Magidor concerning strong compactness, and a problem by Lambie-Hanson and Rinot concerning the $C$-sequence number are solved as well.