Let
L
L
contain only the equality symbol and let
L
+
L^+
be an arbitrary finite symmetric relational language containing
L
L
. Suppose probabilities are defined on finite
L
+
L^+
structures with ‘edge probability’
n
−
α
n^{-\alpha }
. By
T
α
T^{\alpha }
, the almost sure theory of random
L
+
L^+
-structures we mean the collection of
L
+
L^+
-sentences which have limit probability 1.
T
α
T_{\alpha }
denotes the theory of the generic structures for
K
α
\mathbb {K}_{\alpha }
(the collection of finite graphs
G
G
with
δ
α
(
G
)
=
|
G
|
−
α
⋅
|
edges of
G
|
\delta _\alpha (G) = |G| - \alpha \cdot |\text {edges of $G$}|
hereditarily nonnegative).
Theorem.
T
α
T^{\alpha }
, the almost sure theory of random
L
+
L^+
-structures, is the same as the theory
T
α
T_{\alpha }
of the
K
α
\mathbb {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.