Call a graph $G$ $n$-saturated if for every set $A$ of size $n$ of vertices and all $B\subseteq A$ there is a vertex $v\not\in A$ that forms an edge with all $w\in B$ and does not form an edge with any $w\in A\setminus B$.
A countably infinite graph is isomorphic to the random graph iff it is $n$-saturated for all $n$.
Here are the questions:
Is it true that for all $n$ there is a finite graph (of size at least $n$) that is $n$-saturated? If yes, are there reasonable upper and lower bounds on the size of such a graph?
(An $n$-saturated graph of size $\geq n$ has at least $n+2^n$ vertices, but this is probably far from optimal.)
Is every finite graph an induced subgraph of a finite $n$-saturated graph? (Edit: As Ori Gurel-Gurevich pointed out, the second question is silly. Clearly, if $G$ is $n$-saturated (and of size at least $n$), then it has every graph with $n$-vertices as an induced subgraph.)
Example: There is a $2$-saturated graph: Let the vertices of $G$ be the $2$-element subsets of a set with $6$ elements. Two of those vertices form an edge if they (as sets) have a non-empty intersection. It is easily checked that this graph is $2$-saturated. But it also has 30 vertices. That seems a lot.