The Asymptotic Bound of the Lubell Function for Diamond-free Families

2012/06/05 by W. Li, Li, Wei-Tian
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1206.0806

openalex publication_date 2012/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a family of subsets of [n]:=1,2,...,n, the Lubell function is defined as \hb_n(\F):=∑_F∈\F\binomn|F|^-1. In \citeGriLiLu, Griggs, Lu, and the author conjectured that if a family \F of subset of [n] does not contain four distinct sets A, B, C and D forming a diamond, namely A⊂ B∩ C and B∪ C⊂ D, then \hb_n(\F)≤ 2+\lfloor(n²)/(4)\rfloor/(n²-n). Moreover, the upped bound is achieved by three types of families. In this paper, we prove the upper bound in the conjecture is asymptotically correct. In addition, we give some results related to the problem of maximizing the Lubell function for the poset-free families.

Related

• Diamond-free Families
• Poset-free Families and Lubell-boundedness
• An upper bound on the size of diamond-free families of sets
• An improved bound on the diamond-free poset problem
• A note on the size of N-free families