The Rado graph contains every finite graph as an induced subgraph. It surely contains *some* finite graphs infinitely often as an induced subgraph, e.g. $K\_2$. Does it contain *all* finite graphs infinitely often as an induced subgraph? Or can an example of a graph be given that is *not* contained infinitely often?



Rado graph containing infinitely many isomorphic subgraphs 
It must contain every finite subgraph infinitely often
as an induced subgraph. For a finite graph $G$ and the positive integer
$n$ consider the graph $H$ consisting of $n$ vertex-disjoint copies of $G$.
As $H$ is an induced subgraph of Rado then there are $n$ vertex-disjoint
induced subgraphs of Rado isomorphic to $G$.


According to Wikipedia, Rado also has every countable graph
as an induced subgraph (I wasn't aware of this until now). Then
the above argument will work for countable graphs too.


