Two Results on Union-Closed Families

2017/08/04 by Karpas, Ilan · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.01434

Abstract

We show that there is some absolute constant c>0, such that for any union-closed family F ⊆ 2^[n], if |F| ≥ ((1)/(2)-c)2ⁿ, then there is some element i ∈ [n] that appears in at least half of the sets of F. We also show that for any union-closed family F ⊆ 2^[n], the number of sets which are not in F that cover a set in F is at most 2^n-1, and provide examples where the inequality is tight.

Cited by

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

Related

• A constant lower bound for the union-closed sets conjecture
• Further analysis on the second frequency of union-closed set families
• Note on the union-closed sets conjecture
• The union-closed set conjecture is true
• Notes on the Union Closed Sets Conjecture