Some new Bollob\'as-type inequalities

A family of disjoint pairs of finite sets P=\(Aᵢ,Bᵢ) i∈[m]\ is called a Bollob\'as system if Aᵢ∩ Bⱼ≠∅ for every i≠ j, and a skew Bollob\'as system if Aᵢ∩ Bⱼ≠∅ for every $i<j$. Bollob\'as proved that for a Bollob\'as system, the inequality {equation*} ∑ᵢ₌₁^m{|A_i|+|B_i|}{|A_i|}⁻¹≤ 1 {equation*} holds. Heged\"{u}s and Frankl generalized this theorem to skew Bollob\'as systems with the inequality {equation*} ∑ᵢ₌₁^m{|A_i|+|B_i|}{|A_i|}⁻¹≤ 1+n, {equation*} provided Aᵢ,Bᵢ⊆ [n]. In this paper, we improve this inequality to {equation*} ∑ᵢ₌₁^m ((1+|A_i|+|B_i|) {|A_i|+|B_i|}{|A_i|})⁻¹ ≤ 1 {equation*} with probabilistic method. We also generalize this result to partitions of sets on both symmetric and skew cases.