Theory of the random graph ($T_{\mathrm{rg}}$)

The theory of the random graph says that the edge relation is irreflexive and symmetric and that, for every two finite disjoint vertex sets, a new vertex is adjacent to every vertex of the first and none of the second.

Every small partial embedding of a monster random graph extends by back-and-forth to an automorphism, using saturation and the extension axioms. It is therefore elementary, and the theory of the random graph eliminates quantifiers.

For every small set $A$ in a monster random graph,Any vertex outside $A$ has infinitely many conjugates over $A$ with the same adjacency pattern.

The three-types of the random graph are determined by equality and adjacency among three variables. There is one all-equal type, six types with exactly two variables equal, and eight types with all variables distinct, for a total of fifteen.