Diamond-free Families
Jerrold R. Griggs, Wei-Tian Li, Linyuan Lu

TL;DR
This paper investigates the maximum size of families of subsets avoiding certain posets, especially diamond-shaped ones, and makes progress on a longstanding conjecture about their asymptotic density.
Contribution
It introduces the use of the Lubell function to analyze diamond-free families and determines the limit ratio for infinitely many diamond sizes, advancing understanding of the poset extremal problem.
Findings
Determined (\uD_k) for infinitely many k values.
Proved an upper bound of 2 3/11 for (_2).
Progressed towards proving (_2)=2.
Abstract
Given a finite poset P, we consider the largest size La(n,P) of a family of subsets of that contains no subposet P. This problem has been studied intensively in recent years, and it is conjectured that exists for general posets P, and, moreover, it is an integer. For let denote the -diamond poset . We study the average number of times a random full chain meets a -free family, called the Lubell function, and use it for to determine for infinitely many values . A stubborn open problem is to show that ; here we make progress by proving (if it exists).
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph theory and applications
