Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2011 / B / 5 / b / Solution

The symmetric Lovász local lemma states: if each bad event has probability at most $p$, each event is independent of the collection of all but at most $d$ other events, andthen the probability that none of the bad events occurs is positive. Independence is from the sigma-algebra generated by the indicated nonneighbours, as in part (a). A sufficient sharper parameter condition is $p\le x(1-x{)}^d$ for some $x\in (0,1)$; taking $x=1/(d+1)$ gives the displayed standard bound when $d\ge 1$. When $d=0$, mutual independence directly gives the conclusion under the displayed bound.