 Bollobas set-pairs inequality

For disjoint pairs $A_i,B_i$ such that $A_i\cap B_j\ne \varnothing$ whenever $i\ne j$, the displayed sum is at most one. In a uniformly random permutation, the events that all of $A_i$ precedes all of $B_i$ are disjoint events, and their probabilities are the summands. Taking $B_i$ to be the complement of the members $A_i$ of an antichain recovers the LYM inequality.