Is there a unique graph $G$ such that a random graph with vertex set $\mathbb{R}$ and a half chance of any two vertices being connected is isomorphic to $G$ with probability $1$?
I am curious about whether there is an uncountable version of the Rado graph, and the question is equivalent to asking whether there is a graph with vertex set $\mathbb{R}$ that contains as induced subgraphs all other graphs with vertex sets of cardinality up to that of $\mathbb{R}$.