On diamond-free subposets of the Boolean lattice

TL;DR

This paper investigates the maximum size of diamond-free subposets within the Boolean lattice, establishing an upper bound of approximately 2.25 times the middle binomial coefficient and analyzing the Lubell function.

Contribution

It proves a new upper bound on the size of diamond-free families in the Boolean lattice and determines the asymptotic maximum of their Lubell function.

Findings

Maximum size of diamond-free families is at most (2.25+o(1)) times the middle binomial coefficient.

Lubell function of such families is at most 2.25+o(1), asymptotically optimal.

The results improve understanding of forbidden subposet configurations in the Boolean lattice.

Abstract

The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: $A\subset B,C\subset D$. A diamond-free family in the $n$-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements $B$ and $C$ may or may not be related. There is a diamond-free family in the $n$-dimensional Boolean lattice of size $(2-o(1)){n\choose\lfloor n/2\rfloor}$. In this paper, we prove that any diamond-free family in the $n$-dimensional Boolean lattice has size at most $(2.25+o(1)){n\choose\lfloor n/2\rfloor}$. Furthermore, we show that the so-called Lubell function of a diamond-free family in the $n$-dimensional Boolean lattice is at most $2.25+o(1)$, which is asymptotically best possible.