Uncountable random graphs

There is a theorem from Erdos and Renyi that says that a random graph on $\aleph_0$ vertices (where each pair of vertices is connected with probability equal to $\frac{1}{2}$) will be isomorphic to the Rado graph with probability 1. Does an analogous result hold for random graphs on $\kappa$ vertices, for $\kappa > \aleph_0$? The proof I have read of the Erdos-Renyi theorem does not appear to generalize to larger cardinals.

More generally, how does one compute probabilities in cases like this (where, in general, the sample space and the event are both sets of the same cardinality $\lambda > 2^{\aleph_0}$. What sorts of arguments are used to show that such an event has probability $0$ or $1$. If anyone has an example of such an problem and solution, I would appreciate it!

• I don't know references, but Wikipedia says that there are many nonisomorphic uncountable graphs with the universal property. en.m.wikipedia.org/wiki/Random_graph Commented Nov 1, 2015 at 19:12
• I saw that too, but I don't think it answers the question one way or the other (at least not immediately). Commented Nov 1, 2015 at 19:23
• Also, it is not really clear in the article which of the statements is the one that they are saying doesn't generalize to uncountable cardinals. On the face of thing, it is possible that there are multiple isomorphism classes all having the extension property, yet almost all randomly generated graphs lie in just one of those classes. Commented Nov 1, 2015 at 19:30

When people talk about "random graphs on $$\kappa$$ vertices", they just mean graphs with the same first order theory as the random graph (i.e. graphs with the same extension property). Under this definition, I will sketch how to show there are non-isomorphic random graphs of any uncountable cardinality.
First, let's look at random graph completions. Let $$G$$ be any infinite graph. Let $$G'$$ be the graph obtained by, for each pair of finite set $$A$$ of verticies in $$G$$, adjoining a new vertex $$v_A$$ to $$G$$ adjacent to every vertex in $$A$$ (and no other verticies in $$G$$). If we repeat this countably many times, we will get a random graph, $$\overline{G} = G\cup G'\cup G''\cup \dots$$.
Let $$G_1$$ be the graph on $$\kappa$$ with no edges. One can show $$\overline{G_1}$$ has no uncountable cliques (essentially because verticies in $$G'\setminus G$$ have only finitely many neighbors in $$G$$ and no neighbors in $$G'\setminus G$$). Let $$G_2$$ be the complete graph on $$\kappa$$. Clearly, $$\overline{G_2}$$ does have an uncountable clique. So, these are two non-isomorphic random graphs.
I believe one can do a stationary set coding argument (like is often used for linear orders) to show there are exactly $$2^\kappa$$ many random graphs of cardinality $$\kappa$$ up to isomorphism, but I may need to assume things like $$\kappa$$ is regular; I haven't thought too hard about it yet.