RANDOMNESS AND SEMIGENERICITY

Let L contain only the equality symbol and let L+ be an arbitrary fini te symmetric relational language containing L. Suppose probabilities a re defined on finite L+ structures with 'edge probability' n(-alpha). By T-alpha, the almost sure theory of random L+-structures we mean the collection of L+-sentences which have limit probability 1. T-alpha de notes the theory of the generic structures for K-alpha (the collection of finite graphs G with delta(alpha)(G) = \G\ - alpha . \ edges of G \ hereditarily nonnegative). 0.1. Theorem. T-alpha, the almost sure th eory of random L+-structures, is the same as the theory T-alpha of the K-alpha-generic model. This theory is complete, stable, and nearly mo del complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.