### 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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