  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 K_alpha, (the collection of finite graphs G with
delta_{alpha}(G)=|G|- alpha. | edges of G | hereditarily nonnegative.)
  THEOREM: T_alpha, the almost sure theory 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 model complete. Moreover, it has the finite model property
and has only infinite models so is not finitely axiomatizable.
