Random Graphs correlation inequalities

Question

Is there anywhere I can find a proof of the first inequality that $$P(B_{i}| \bigcap_{j \in J} \bar B_{j}) \leq P(B_{i})$$ It is on page $13$ and is the first inequality presented at the start of section $2$ on these notes (Wayback Machine).

Answer 1

Using the terminology from the document. The event $A_i \subseteq R$ is increasing (i.e adding points to $R$ does not change the inclusion), and the event $A_j \not\subseteq R$ is decreasing. Also, intersection of decreasing events is still decreasing, and then you can apply Kleitman's lemma or FKG inequality which will give you that $B_i$ and $\cap_{j \in J} \bar B_{j}$ are negatively correlated.