An improved bound on the diamond-free poset problem

TL;DR

This paper establishes a tighter upper bound on the size of diamond-free families in the Boolean lattice, advancing understanding of extremal poset problems with implications for combinatorics.

Contribution

The paper provides an improved upper bound on the maximum size of diamond-free families in the Boolean lattice, refining previous bounds and contributing to extremal poset theory.

Findings

New bound: || leq 2.206653 * binomial coefficient

Improved previous bounds on diamond-free poset families

Advances theoretical understanding of extremal set families

Abstract

In the theory of partially-ordered sets, the two-dimensional Boolean lattice is known as the diamond. In this paper, we show that, if $\mathcal{F}$ is a family in the $n$-dimensional Boolean lattice that has no diamond as a subposet, then $|\mathcal{F}|\leq 2.206653{n\choose \lfloor n/2\rfloor}$, improving a bound by the authors and Michael Young.