Variations on the Bollobás set-pair theorem

Abstract

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 $A_i\cap B_j\ne \varnothing$ for all $1\le i<j\le m$. The best possible inequality $$\sum _{i=1}^m\frac{1}{\left(\genfrac{}{}{0ex}{}{|A_i|+|B_i|}{|A_i|}\right)}\le n+1.$$ is established along with some more results of similar flavor.

12 pages