Historical motivations. Modern model theory started when M. Morley [Mor65] proved his famous theorem on the categoricity in any uncountable cardinal of first order theories categorical in one uncountable cardinal. He introduced for that purpose an ordinal valued rank on types of such a theory, later shown to be finite by J. Baldwin [Bal73].

fact 1.1. An uncountably categorical first order theory has finite Morley rank.

This was the begining of the fantastic development of the classification theory by S. Shelah on the number of non-isomorphic uncountable models of a first order theory [She90], and more precisely the developments of stability theory with all subsequent generalizations of linear or algebraic independance in classical mathematical structures such as vector spaces or fields. In the meantime, it appeared interesting to study structures with such a given model-theoretic property. The first result of this kind was obtained by A. Macintyre [Mac71].

macintyre's theorem.An infinite field of finite Morley rank is algebraically closed.

On the other hand, B. Zilber showed in his work on uncountably categorical structures the following result on simple groups of finite Morley rank [Zil77].

fact 1.2. An infinite simple group of finite Morley rank is uncountably categorical.

Naturally, this result gave the feeling that simplicity was certainly hiding stronger structural properties. Hence, motivated by a sense that most structures already exist “in nature”, G. Cherlin and B. Zilber proposed independently the following conjecture in the late seventies.