We give an axiomatic framework for the non-modular simple 0-categorical structures constructed by Hrushovski. This allows us to verify some of their properties in a uniform way, and to show that these properties are preserved by iterations of the construction.

For $n < \omega$ , expand the structure (n, S, I, F) (with S the successor relation, I, F as the initial and final element) by forming graphs with edge probability n-α for irrational α, with $0 < \alpha < 1$ . The sentences in the expanded language, which have limit probability 1, form a complete and stable theory.