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__Ph D thesis - abstract__

We consider complete theories, in a first order language, with infinite
models. A model of such a theory is said to be* weakly Ehrenfeucht*
if it can be expanded to the algebraic closure of an indiscernible sequence,
and is said to be *strongly Ehrenfeucht* if, in addition, it is saturated
in a power which is strictly greater than the cardinality of the expanded
language. Given the conflicting combinatorial requirements of these two
properties, one naturally asks whether strongly Ehrenfeucht models exist.
One also may look for a characterization of theories with strongly Ehrenfeucht
models.
In chapters 2 and 3, we show, among other things that: if a theory has
a strongly Ehrenfeucht model then it has arbitrarily saturated strongly
Ehrenfeucht models; an unstable theory having strong Ehrenfeucht models
is not multi-ordered and thus does not have the independence property;
the stable theories having strongly Ehrenfeucht models are exactly the
superstable theories and every saturated model of a superstable theory
is strongly Ehrenfeucht. As well as using properties of indiscernible sequences,
we also use some Stability Theory, various combinatorial tools and the
model theory of orders - in particular dense orders without endpoints.

In chapter 4, we consider a strongly minimal theory (note that each
infinite-dimensional model is a strongly Ehrenfeucht) and we study extensions
of this theory in which the algebraic and definable closures coincide;
such extensions are called *multiplicity-1 extensions*. We show that
the stability spectrum of multiplicity-1 extensions is affected by the
combinatorial properties of the theory: if the theory is not unimodular,
its multiplicity-1 extensions are not totally transcendental; if the theory
strongly eliminates imaginaries, its multiplicity-1 extensions are not
stable.

REFERENCES

A. EHRENFEUCHT & A. MOSTOWSKI,
Models of axiomatic theories admitting automorphisms. *Fundamenta Mathematicae*
43 (1956) 50-68

R. KAYE, Indiscernibles, in *Automorphisms of
first-order structures*, pp 257-279. Clarendon Press, Oxford 1994

D. LASCAR, Autour de la propriété
du petit indice. *Proceedings of the London Mathematical Society*
62 (1991) 25-53

J.P. RESSAYRE, Introduction aux modèles
récursivement saturés, in *Séminaire général
de logique 1983-84*, pp 53-72. Publications Mathématiques de
l'Université Paris VII

J.H. SCHMERL, Recursively saturated models generated
by indiscernibles. *Notre Dame Journal of Formal Logic* 26 (1985)
99-105

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