Suppose you have a graph, say a geometric random graph, with $n$ nodes where each link appears with probability $p$. Assuming $E_k$ is the event that the graph has minimum degree $k$, is it possible to find a lower-bound of the form
$$P(E_k) \geq [f(k)]^n \> ?$$
In other words, I'd like to find an expression that allows me to treat the vertices as if they were independent (which is not true).
Perhaps that would be helpful: The Maximum Degree of a Random Graph. Note that minimum and maximum degree are the same (just take $1-p$ instead of $p$).