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

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