Variations on the Bollob\'as set-pair theorem

Let $X$ be an $n$-element set. A set-pair system \mbox{\cal P}=\{(A_i,B_i)\}_{1\leq i\leq m} is a collection of pairs of disjoint subsets of $X$. It is called skew Bollob\'as system if $A_i\cap B_j\neq \emptyset$ for all $1\leq i<j \leq m$. The best possible inequality $$\sum_{i=1}^m \frac{1}{{|A_i|+|B_i| \choose |A_i|}}\leq n+1.$$ is established along with some more results of similar flavor.