Correlation graph of events

For bad events $A_i$, a correlation graph is sufficient for the Lopsided Lovász local lemma when, for every set $S$ of nonneighbors and every positive-probability conditioning event, $\mathbb{P}(A_i\mid {\bigcap }_{j\in S}{A}_j^c)\le \mathbb{P}(A_i)$. A dependency graph of events satisfies this with equality. The condition concerns simultaneous avoidance, not just signs of pairwise correlations.