Maximal failures of sequence locality in a.e.c.

by Shelah. [Sh:932]

We are interested in examples of a.e.c. with amalgamation having some (extreme) behaviour concerning types. Note we deal with k being sequence-local, i.e. local for increasing chains of length a regular cardinal. For any cardinal theta >= aleph_0 we construct an a.e.c. with amalgamation k with L.S.T. (k) = theta,| tau_K | = theta such that {kappa : kappa is a regular cardinal and 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. We also prove a similar result to ``(2^kappa, kappa)-compact for types''.

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