On Local Club Condensation

TL;DR

This paper explores local club condensation in extender models, establishing its connection with subcompact cardinals and demonstrating how to force models where local club condensation and related combinatorial principles hold across intervals of ordinals.

Contribution

It characterizes local club condensation in extender models, links it to subcompact cardinals, and shows how to force models satisfying local club condensation and combinatorial principles.

Findings

Equivalence between failure of local club condensation and subcompact cardinals.

Construction of models where local club condensation holds for all ordinals in an interval.

Forcing techniques preserving GCH, cardinals, and cofinalities to achieve desired condensation properties.

Abstract

We obtain results on the condensation principle called local club condensation. We prove that in extender models an equivalence between the failure of local club condensation and subcompact cardinals holds. This gives a characterization of $\square_{\kappa}$ in terms of local club condensation in extender models. Assuming $\gch$, given an interval of ordinals $I$ we verify that iterating the forcing defined by Holy-Welch-Wu, we can preserve $\gch$, cardinals and cofinalities and obtain a model where local club condensation holds for every ordinal in $I$ modulo those ordinals which cardinality is a singular cardinal. We prove that if $\kappa$ is a regular cardinal in an interval $I$, the above iteration provides enough condensation for the combinatorial principle $\Dl_{S}^{*}(\Pi^{1}_{2})$, and in particular $\diamondsuit(S)$, to hold for any stationary $S \subseteq \kappa$.