Knaster and friends II: The C-sequence number

TL;DR

This paper introduces the C-sequence number, a new measure of compactness for regular uncountable cardinals, exploring its properties, relationships with large cardinals, and connections to coloring principles.

Contribution

It defines the C-sequence number, establishes its fundamental properties, and links it to large cardinals, stationary reflection, and coloring principles, expanding the understanding of cardinal compactness.

Findings

C-sequence number characterizes compactness of regular uncountable cardinals.

Results include ZFC and independence results relating to large cardinals and reflection.

Connections between C-sequence spectrum and strong coloring principles are established.

Abstract

Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong coloring principle U(...), introduced in Part I of this series.