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' $n^{-\alpha }$. By $T^{\alpha }$, the almost sure theory of random $L^+$-structures we mean the collection of $L^+$-sentences which have limit probability 1. $T_{\alpha }$ denotes the theory of the generic structures for ${\mathbf {K}} _{\alpha }$ (the collection of finite graphs $G$ with $\delta _{\alpha }(G) =|G| - \alpha \cdot |\text { edges of $G$ }|$ hereditarily nonnegative).

. $T^{\alpha }$, the almost sure theory of random $L^+$-structures, is the same as the theory $T_{\alpha }$ of the ${\mathbf {K}} _{\alpha }$-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.