Asymmetric Lovász local lemma
= Asymmetric Lovász local lemma
For a finite family with a <dependency graph of events>, if $0\le x_i<1$ satisfy $\mathbb P(E_i)\le x_i\prod_{j\in N(i)}(1-x_j)$, then $\mathbb P(\bigcap_iE_i^c)\ge\prod_i(1-x_i)>0$. Choosing equal $x_i$ gives the familiar symmetric criterion $eq(d+1)\le1$ when event probabilities are at most $q$ and the maximum dependency degree is $d$.