  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.
