Thresholds and Expectation Thresholds

We consider relations between thresholds for monotone set properties and simple lower bounds for such thresholds. A motivating example (Conjecture 2): Given an n-vertex graph H, write pE for the least p such that, for each subgraph H' of H, the expected number of copies of H' in G=G(n, p) is at least 1, and pc for that p for which the probability that G contains (a copy of) H is 1/2. Then (conjecture) pc=O(pElog n). Possible connections with discrete isoperimetry are also discussed.

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