Some new Bollob\'as-type inequalities

TL;DR

This paper introduces a new Bollobás-type inequality using probabilistic methods, improving previous bounds for disjoint set systems and extending results to set partitions in symmetric and skew cases.

Contribution

It presents a tighter inequality for Bollobás systems, generalizes to partitions, and employs probabilistic techniques for the proofs.

Findings

Improved inequality for Bollobás systems using probabilistic methods

Generalization to set partitions in symmetric and skew cases

Enhanced bounds compared to previous theorems

Abstract

A family of disjoint pairs of finite sets $\mathcal{P}=\{(A_i,B_i)\mid i\in[m]\}$ is called a Bollob\'as system if $A_i\cap B_j\neq\emptyset$ for every $i\neq j$, and a skew Bollob\'as system if $A_i\cap B_j\neq\emptyset$ for every $i<j$. Bollob\'as proved that for a Bollob\'as system, the inequality \begin{equation*} \sum_{i=1}^m\binom{|A_i|+|B_i|}{|A_i|}^{-1}\leq 1 \end{equation*} holds. Heged\"{u}s and Frankl generalized this theorem to skew Bollob\'as systems with the inequality \begin{equation*} \sum_{i=1}^m\binom{|A_i|+|B_i|}{|A_i|}^{-1}\leq 1+n, \end{equation*} provided $A_i,B_i\subseteq [n]$. In this paper, we improve this inequality to \begin{equation*} \sum_{i=1}^m \left((1+|A_i|+|B_i|) \binom{|A_i|+|B_i|}{|A_i|}\right)^{-1} \leq 1 \end{equation*} with probabilistic method. We also generalize this result to partitions of sets on both symmetric and skew cases.