Short proofs for generalizations of the Lovász Local Lemma: Shearer's condition and cluster expansion

The Lovász Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Over the years, more general conditions have been discovered under which the conclusion of the lemma continues to hold. In this note we provide short proofs of two of those more general results: Shearer's lemma and the cluster expansion lemma, in their "lopsided" form. We conclude by using the cluster expansion lemma to prove that the symmetric form of the local lemma holds with probabilities bounded by $1/ed$, rather than the bound $1/e(d+1)$ required by the traditional proofs.