Model Theory, Volume 47, Number 11

group Aut(M). Stability and Diophantine Geometry A structure M is said to be unstable if it interprets a bipartite graph (P,Q,R) with the feature that for each n there are ai ∈ P and bi ∈ Q for i = 1, . . . , n such that R(ai, bj ) if and only if i < j. A complete theory is unstable if some (any) model is unstable. If M is unstable (witnessed by (P,Q,R) ) and saturated, then there are ai and bi for i = 1,2, . . . such that R(ai, bj ) if i < j. A structure or theory is stable if it is not unstable. By definition stability is an invariant of the bi-interpretability type. The connection between stability and IT is: if T is unstable, then IT (κ) = 2κ (the maximum possible) for all uncountable cardinals κ. So in the context of classifying the possible functions IT , it was natural to focus on stable theories. Stability is a very strong property. There are few natural examples of stable structures: abelian groups (G,+), algebraically closed and separably closed fields (K,+, · ), differentially closed fields (K,+, · ,D) . More recently it was realized that compact complex manifolds are also stable; the structure on the compact complex manifold X consists of the analytic subvarieties of X, X ×X,.... On the other hand, typically the structures considered in earlier sections, such as the real field and p-adic field, are