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).