Comparison of Two Convergence Criteria for the Variable-Assignment Lopsided Lovasz Local Lemma

The Lopsided Lovász Local Lemma (LLLL) is a probabilistic tool which cornerstone of the method combinatorics, shows that it possible to avoid collection "bad" events as long their probabilities and interdependencies are sufficiently small. strongest criterion can be stated in these terms due Shearer (1985), although technically difficult apply constructions combinatorics.
 
 original formulation LLLL was non-constructive; seminal algorithm Moser & Tardos (2010) gave an efficient constructive for nearly all applications it, including $k$-SAT instances with bounded number occurrences per variables. Harris (2015) later alternate this converge, appeared give stronger bounds than standard LLL. Unlike LLL or its variants, depends fundamental way on decomposition bad-events into variables.
 In note, we show given by some cases even Shearer's criterion. We construct formulas variable occurrence, satisfied while violated. fact, there exponentially growing gap between provable from any form bound shown Harris.