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In the Erdös-Rényi model for random graphs there is a lot of results stating sharp phase transitions for the probability of a random graph to contain a fixed prescribed subgraph.

Is there a model of Erdös-Renyi graphs, which do contain a fixed subgraph?

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    $\begingroup$ This is probably not exactly what you are interested in, but there is an interesting theory of conditioning an ER random graph on containing many more than its expected number of some prescribed subgraph, e.g., triangles: yufeizhao.wordpress.com/2014/02/25/upper-tail $\endgroup$
    – Sam Hopkins
    Commented Jun 22, 2017 at 20:29
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    $\begingroup$ Could you possibly say a little more about what you're after? For me the key feature of the Erdős–Rényi model is the independence of edges, which doesn't play nicely with most additional conditions. $\endgroup$
    – Ben Barber
    Commented Jun 22, 2017 at 21:15

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