### Dependent $T$ and Existence of limit models

by Shelah. [Sh:877]

Tbilisi Math J, 2014

We continue [Sh:868] and [Sh:783]. The problem there is when
does (first order) T have a model M of cardinality lambda
which is (one of the variants of) a limit model for cofinality
kappa, and the most natural case to try is lambda = lambda^{<
lambda}> kappa = cf (kappa)>|T| . The stable theories has one;
are there unstable T whnce of limit models
AUTHORS: Saharon Shelah
ich has such limit models? We find one:
the theory T_ord of dense linear orders. So does this hold
for all unstable T ? As T_ord is prototypical of dependent
theories, it is natural to look for independent theories. A strong,
explicit version of T being independent is having the strong
independence property. We prove that for such T there are no
limit models. We work harder to prove this for every dependent
T, i.e., with the independence property though a weaker version. This
makes us conjecture that any dependent T has such models. Toward
this end we continue the investigation of types for dependent T .

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