Diamond-free Families

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 $[n]:=\{1,...,n\}$ that contains no subposet P. This problem has been studied intensively in recent years, and it is conjectured that $\pi(P):= \lim_{n\rightarrow\infty} La(n,P)/{n choose n/2}$ exists for general posets P, and, moreover, it is an integer. For $k\ge2$ let $\D_k$ denote the $k$-diamond poset $\{A< B_1,...,B_k < C\}$. We study the average number of times a random full chain meets a $P$-free family, called the Lubell function, and use it for $P=\D_k$ to determine $\pi(\D_k)$ for infinitely many values $k$. A stubborn open problem is to show that $\pi(\D_2)=2$; here we make progress by proving $\pi(\D_2)\le 2 3/11$ (if it exists).