On the probability of nonexistence in binomial subsets

Given a hypergraph Γ = (Ω, X) and a sequence p = (p_ω)_ω∈Ω of values in (0, 1), let Ω_p be the random subset of Ω obtained by keeping every vertex ω independently with probability p_ω. We investigate the general question of deriving fine (asymptotic) estimates for the probability that Ω_p is an independent set in Γ, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of upper and lower bounds on this probability, each of which can be evaluated explicitly in terms of the joint cumulants of small sets of edge indicator random variables. Under certain natural conditions, these upper and lower bounds coincide asymptotically, thus giving the precise asymptotics of the probability in question. We demonstrate the applicability of our results with two concrete examples: subgraph containment in random (hyper)graphs and arithmetic progressions in random subsets of the integers.