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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?

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up vote 9 down vote accepted

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.

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My god, things are so simple sometimes. Thanks! – Hans Stricker Mar 17 '10 at 10:55
Sometimes simple things can be hard to see :-) – Robin Chapman Mar 17 '10 at 21:02

I realise that the question is almost three years old, but maybe that's not that long in Maths.

I am not sure who the following is originally due to, but as far as I am aware, it's the standard to show that the Rado graph contains every countable graph. The idea is to start with whatever graph you want and construct the Rado graph around it.

Let $G=G_0$ be any countable graph. For $i>0$ define a new Graph $G_i$ as follows:

  • The vertices $V(G_{i})$ of $G_i$ are all of $V(G_{i-1})$ plus an extra vertex $v_A$ for every finite subset $A$ of $V(G_{i-1})$.

  • All edges of $V(G_{i-1})$ are also edges of $V(G_{i})$

  • Add an edge between $a\in V(G_{i-1})$ and $v_{A}\in V(G_{i})$ whenever $a\in A$.

Let $R$ be the union of the graphs $G_i$ over all $i>0$. Then show that $R$ is the Rado graph. Clearly, $R$ contains $G=G_0$.

One of the nice things about this construction is that you can show without much difficulty that any automorphism of $G$ extends to an automorphism of $R$. So not only does the Rado graph contain every countable graph $G$, but it contains "special" copies of $G$ with the above extension property.

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