A sharper local lemma with improved applications

Abstract

We give a new family of Lovász Local Lemmas (LLL), with applications. Shearer has given the most general condition under which the LLL holds, but the original condition of Lovász is simpler and more practical. Do we have to make a choice between practical and optimal? In this article we present a continuum of LLLs between the original and Shearer's conditions. One of these, which we call Clique LLL (CLLL), particularly stands out, and is natural in those settings, where the event space is defined with discrete independent random variables (á la Moser and Tardos ). Using this version we get improved bounds in applications for Acyclic Edge Coloring and Non-repetitive Vertex Coloring. © 2012 Springer-Verlag.