Lecture 14 : The Lovász Local Lemma and the Independence Polynomial

iff px ∈ [0, 1) for all x. When the events are not independent, an answer is given by the Lovász local lemma (and its variations). Definition 14.1. We say that G is the dependency graph of (Ax)x∈X if for all x ∈ X, Ax is independent of the σ-algebra generated by the collection {Ay | y / ∈ Γ∗(x)} (where Γ(x) is the set of neighbours of x in G and Γ∗(x) := Γ(x) ∪ {x}). Theorem 14.2 (Lovász local lemma). Let G be the dependency graph for the of events (Ax)x∈X , and suppose that (rx)x∈X ∈ [0, 1)X such that, for each x, P(Ax) ≤ rx ∏