The countable random graph, also known as the Rado graph, is characterized as the unique countable graph in which every two disjoint finite sets $A$ and $B$ of vertices admit a vertex $p$ connected to every vertex in $A$ and to none in $B$. This is a kind of saturation property, and any two countable graphs exhibiting it are isomorphic by a fun back-and-forth construction; indeed, any finite partial isomorphism extends in this situation to a full isomorphism. The same argument shows that every countable graph has many copies inside the Rado graph. The Rado graph can also be described as the almost-sure result of constructing an edge between any two distinct integers with independent probability $p\in(0,1)$, essentially because the probability of fulfilling the property for any two $A$ and $B$ is one, since there are infinitely many independent chances to get the pattern correct. (Note, in this question, graphs have no trivial loops or parallel edges.)
My questions are:
Is the Rado graph a Cayley graph? That is, can one orient and label the edges with generators in such a way that the Rado graph is the resulting Cayley graph of the corresponding group presentation? (I expect that the answer is yes, by building up a simply transitive group action and employing Sabidussi's theorem, but I would be interested to see an elegant account of this idea if it works.)
If so, is the resulting group unique? What is its nature?
For example, can one make the Rado graph a Cayley graph of a group with torsion? without torsion?
Does the group exhibit a probabilistic account, as the graph does?
If the Rado graph is a Cayley graph, what universal properties does the resulting group enjoy similar to the attractive universal properties of the graph? For example, every countable Cayley graph finds a copy in the Rado graph; can we always find a copy whose Cayley labeling extends to a Cayley labeling of the full Rado graph?
It seems that the ideas of the random graph extend in several ways to the context of directed graphs. For example, one could ask that for every disjoint triple of sets $A$, $B$ and $C$, there is a node $p$ pointing at all nodes in $A$, pointed at by all nodes in $B$ and with no edges between $p$ and nodes in $C$. This saturation property again characterizes the directed graph up to isomorphism by a back-and-forth construction. I suppose this graph would also be the almost-sure result of building the Rado graph probabalistically, but also orienting the edges with independent probability $p$.
Are any of the questions above affected by this change? That is, in this version of the questions, the edge orientations are already given, but satisfy the saturation property.
And what of the uncountable analogues of the random graph? For example, if the Continuum Hypothesis holds, then there is a (unique) graph of size $\aleph_1$ satisfying the saturation property with respect to all countable disjoint subsets $A$, $B$, that is, having a vertex connected to all vertices in $A$ and to none in $B$. Is this graph a Cayley graph, and so on?