MTH 742 P : Advanced Combinatorics 12 / 12 / 2016 Lecture 11 : Lovász Local Lemma

Usually, when we use the probabilistic method, the desired event holds with very high probability. On the other hand, suppose we have n mutually independent events, each of which holds with probability p, then the conjunction of all of them holds with probability p. This is exponentially small but strictly positive. In most applications, the desired event cannot be decomposed into n mutually independent ones. The Lovász Local Lemma provides a way to deal with dependencies. It was first found by Erdős and Lovász in 1975 and is extremely powerful, especially if the dependencies are rare. Let A1, A2, . . . , An be events in an arbitrary probability space. A graph D = (V,E) on the set of vertices V = [n] is called the dependency graph for the events A1,. . . ,An if for each i ∈ [n], the event Ai is mutually independent of all events that are not in Γ(i) := Γ(i)∪{i}, where Γ(i) = {j : (i, j) ∈ E}.