Variations on the Bollobás set-pair theorem

Let $X$ be an $n$-element set. A set-pair system $P={\left\{(A_i,B_i)\right\}}_{1\le i\le m}$ is a collection of pairs of disjoint subsets of $X$. It is called skew Bollobás system if $A_i\cap B_j\ne 0̸$ for all $1\le i<j\le m$. The best possible inequality $\sum _{i=1}^m\frac{1}{\left(\frac{\left|A_i\right|+\left|B_i\right|}{\left|A_i\right|}\right)}\le n+1.$ is established along with some more results of similar flavor.