Randomness and semigenericity

Let L contain only the equality symbol and let L⁺ be an arbitrary finite symmetric relational language containing L. Suppose probabilities are defined on finite L⁺ structures with 'edge probability' n-α. By Tα, the almost sure theory of random L⁺-structures we mean the collection of L⁺-sentences which have limit probability 1. T_α denotes the theory of the generic structures for K_α (the collection of finite graphs G with 6_α(G) = \G\ -α \edges of G \ hereditarily nonnegative). 0.1. Theorem. T, the almost sure theory of random L+ -structures, is the same as the theory T_α of the K_α -generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.