Cofinality spectrum problems: the axiomatic approach

by Malliaris and Shelah. [MiSh:1070]
Topology and its Applications, 2016
Let X be a set of definable linear or partial orders in some model. We say that X has emph {uniqueness} below some cardinal {t}_* if for any regular kappa < {t}_*, any two increasing kappa-indexed sequences in any two orders of X

have the same co-initiality. Motivated by recent work, we investigate this phenomenon from several interrelated points of view. We define the lower-cofinality spectrum for a regular ultrafilter {D} on lambda and show that this spectrum may consist of a strict initial segment of cardinals below lambda and also that it may finitely alternate. We connect these investigations to a question of Dow on the conjecturally nonempty (in ZFC) region of OK but not good ultrafilters, by introducing the study of so-called `automorphic ultrafilters' and proving that the ultrafilters which are automorphic for some, equivalently every, unstable theory are precisely the good ultrafilters. Finally, we axiomatize a general context of ``lower cofinality spectrum problems'', a bare-bones framework consisting essentially of a single tree projecting onto two linear orders. We prove existence of a lower cofinality function in this context show that this framework holds of theories which are substantially less complicated than Peano arithmetic, the natural home of cofinality spectrum problems. Along the way we give new analogues of several open problems.

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