The infamous upper tail

Let Γ be a finite index set and k ≥ 1 a given integer. Let further S ⊆ [Γ]≤k be an arbitrary family of k element subsets of Γ. Consider a (binomial) random subset Γp of Γ, where p = (pi : i ∈ Γ) and a random variable X counting the elements of S that are contained in this random subset. In this paper we survey techniques of obtaining upper bounds on the upper tail probabilities P(X ≥ λ + t) for t > 0. Seven techniques, ranging from Azuma’s inequality to the purely combinatorial deletion method, are described, illustrated and compared against each other for a couple of typical applications. As one application, we obtain essentially optimal bounds for the upper tails for the numbers of subgraphs isomorphic to K4 or C4 in a random graph G(n, p), for certain ranges of p.