Let T be a countable, small simple theory. In this paper, we prove that for such T, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.

§1. Introduction. In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular (...) algebraic structures which have been studied recently turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory T has 2λ many models in any cardinal λ ≥ ω1 + |T|. (shrink)

In this paper a class of simple theories, called the low theories is developed, and the following is proved. Theorem. Let T be a low theory. A set and a, b elements realizing the same strong type over A. Then, a and b realized the same Lascar strong type over A.