ENGLISH ABSTRACT: The definable (p, q)-conjecture is a model-theoretic version of a (p, q)-
theorem in combinatorics, which was expressed in the form of a question
by A. Chernikov and P. Simon in 2015. Researchers have proved that the
property holds for certain classes of structures. Based on those existing results,
the main objective of the present thesis is to show that the definable
(p, q)-conjecture holds for a dense pair of geometric distal structures that
satisfies the following condition: algebraic closure and definable closure
are the same in sense of the original geometric structure. Independently,
we also explore a different approach to prove that under some conditions,
the definable (p, q)-conjecture holds in certain cases for dense pairs of real
closed ordered fields.

(Stellenbosch : Stellenbosch University, 2017-03) Rakotonarivo, Tsinjo Odilon; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences

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ENGLISH ABSTRACT : Imaginaries are definable equivalence classes, which play an important
role in model theory. In this thesis, we are interested in imaginaries of
dense pairs of real-closed fields. More precisely, we consider the following
problem: is acleq equal to dcleq in dense pairs of real-closed fields?
To answer this question, we first present some results about real-closed
fields, which are basically completeness, quantifier elimination and elimination
of imaginaries. Then, we concentrate on the completeness and near
model-completeness for the theory of dense pairs of real-closed fields. And
finally, we present the key point of the thesis. Namely, we demonstrate that
acleq(∅) = dcleq(∅) but there exists A such that acleq(A) 6= dcleq(A)