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.
Categories: math.CO