Variations on the Bollobás 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 Bollobás system if $A_i\cap B_j\mathrm{\ne }\mathrm{\varnothing }$ for all $1\le i<j\le m$. The best possible inequality $$\sum _{i=1}^m\frac{1}{\left(\genfrac{}{}{0px}{}{\mathrm{\mid }{A}_i\mathrm{\mid }+\mathrm{\mid }{B}_i\mathrm{\mid }}{\mathrm{\mid }{A}_i\mathrm{\mid }}\right)}\le n+1.$$ is established along with some more results of similar flavor.