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 Γ p of Γ, where p = ( p i : 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 K 4 or C 4 in a random graph G ( n , p ), for certain ranges of p .