If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 p G called an infinite random graph. Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property:

It turns out that if the vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.




Graph Theory Probability Graph Theory