Some new Bollob\'as-type inequalities

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.