Supersaturation in union-closed families of sets

2026/07/29 by Christopher Bouchard
Mathematics · #math.CO

paper · pdf

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

Let k and n be positive integers such that 2 ≤ k ≤ n+1. We prove that the number of k-chains in a union-closed family with universe [n] and size m is minimized when its member sets are largest possible. We also show that, whenever the minimum is nonzero and m>n, there are no other minimizing families.

Citations

• An upper bound for union-closed family size
• A constant lower bound for the union-closed sets conjecture
• Subsets of posets minimising the number of chains
• Kleitman's conjecture about families of given size minimizing the number of k-chains
• The journey of the union-closed sets conjecture

Related

• Minimizing the Number of Unions
• Chain Conditions and Optimal Elements in Generalized Union-Closed Families of Sets
• An upper bound for union-closed family size
• Multiply union families in ℕⁿ
• An asymptotic version of the union-closed sets conjecture