A lemma on a finite union-closed family of finite sets and its applications

2025/07/15 by Zechun Hu, Hu, Ze-Chun, Yun Q. Shi +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2507.11008

openalex publication_date 2025/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that \mathscrF is a finite union-closed family of sets with ∪_A∈ \mathscrFA=\1,2,…,m\ and m≥ 2. Fix i∈ \1,2,…,m\ and denote \mathscrG:=\A\backslash \i\: A∈ \mathscrF\. For j∈ \1,2,…,m\\backslash\i\, let \mathscrG_j:=\A∈\mathscrG: j∈ A\ and \mathscrF_j:=\A∈\mathscrF: j∈ A\. In this note, we will prove a lemma which says that if \frac|\mathscrG_j||\mathscrG|≥ c (c∈ (0,1]), then \frac|\mathscrF_j||\mathscrF|≥ (1)/(1+2(1-c)/c). Several applications of this lemma will be given.

Citations

• Frequent elements in union-closed set families
• Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling
• The number of abundant elements in union-closed families without small sets
• Notes on the Union Closed Sets Conjecture
• The 6-element case of S-Frankl conjecture (I)
• Two Stronger Versions of the Union-closed Sets Conjecture
• Two Results on Union-Closed Families
• On the Union-Closed Sets Conjecture
• The journey of the union-closed sets conjecture

Related

• The Union-Closed Sets Conjecture for Small Families
• Union-Closed vs Upward-Closed Families of Finite Sets
• Notes on the Union Closed Sets Conjecture
• The union-closed set conjecture is true
• Graph generated union-closed families of sets