Sh:932

• Shelah, S. Maximal failures of sequence locality in a.e.c. Preprint. arXiv: 0903.3614
• Abstract:
  We are interested in examples of AECs \mathfrak{k} with amalgamation having some (extreme) behavior concerning types. Note, we deal with {\mathfrak k} being sequence-local, i.e. local for increasing chains of length a regular cardinal (for types, equality of all restrictions implies equality, some call it tame). For any cardinal \theta \ge \aleph_0 we construct an AEC \mathfrak{k} with amalgamation and {\rm LST}({\mathfrak{k} }) = \theta,|\tau_{\mathfrak K}| = \theta such that \{\kappa:\kappa is a regular cardinal and {\mathfrak K} is not (2^\kappa,\kappa)-sequence-local\} is maximal. In fact, we have a direct characterization of this class of cardinals: the regular \kappa such that there is no uniform \kappa^+-complete ultrafilter (on any \lambda > \kappa). We also prove a similar result to “(2^\kappa,\kappa)-compact for types".
• Version 2025-12-05 (22p)

@article{Sh:932,
 author = {Shelah, Saharon},
 title = {{Maximal failures of sequence locality in a.e.c.}},
 note = {\href{https://arxiv.org/abs/0903.3614}{arXiv: 0903.3614}},
 arxiv_number = {0903.3614}
}