I've found myself needing to prove a certain graph theoretic property has a probabilistic threshold in the Erdos-Renyi model. The problem is that this property is not edge monotone in general. It is, however, edge monotone provided that the graph is 3-connected. My question then is whether there is any probabilistic justification for essentially "chaining" the two thresholds. E.g., once p is at least this large I will be 3-connected almost certainly, at which point my property is monotone and so after specializing p once more I obtain a threshold for my property. If such an argument can be made rigorous, would I be able to immediately calculate what the overall threshold is provided the thresholds on each of the steps?
Any help would be appreciated.
