Some new Bollob\'as-type inequalities
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 …