Some new Bollobás-type inequalities

Abstract

A family of disjoint pairs of finite sets $\mathcal{P}=\{(A_i,B_i)\mid i\in [m]\}$ is called a Bollobás system if $A_i\cap B_j\ne \varnothing$ for every $i\ne j$, and a skew Bollobás system if $A_i\cap B_j\ne \varnothing$ for every $i<j$. Bollobás proved that for a Bollobás system, the inequality \begin{equation*} \sum_{i=1}^m\binom{|A_i|+|B_i|}{|A_i|}^{-1}\leq 1 \end{equation*} holds. Hegedüs and Frankl generalized this theorem to skew Bollobás 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.

10 pages