Variations on the Bollob\'as set-pair theorem

TL;DR

This paper explores variations of the Bollobás set-pair theorem, establishing new inequalities for skew Bollobás systems involving disjoint set pairs and their intersection properties.

Contribution

It introduces and proves new bounds and inequalities for skew Bollobás systems, extending classical combinatorial set-pair theorems.

Findings

Established the optimal inequality for skew Bollobás systems

Derived new bounds for set-pair systems with intersection constraints

Extended classical results to more general set configurations

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 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.