Three layer $Q_2$-free families in the Boolean lattice

TL;DR

This paper determines an improved upper bound on the size of the largest $Q_2$-free family of subsets in the Boolean lattice, specifically those with at most three different set sizes, refining previous bounds.

Contribution

It establishes a tighter upper bound on the maximum size of $Q_2$-free families with limited set sizes, advancing combinatorial understanding.

Findings

Largest $Q_2$-free family size is approximately 2.1547 times N.

Improves previous bound of 2.207N.

Provides exact asymptotic upper bound for such families.

Abstract

We prove that the largest $Q_2$-free family of subsets of $[n]$ which contains sets of at most three different sizes has at most $(3 + 2\sqrt {3})N/3 + o(N) \approx 2.1547N + o(N)$ members, where $N = {n \choose {\lfloor n/2 \rfloor}}$. This improves an earlier bound of $2.207N + o(N)$ by Axenovich, Manske, and Martin.