$Q_2$-free families in the Boolean lattice

TL;DR

This paper determines bounds on the maximum size of families of subsets of [n] that avoid a specific poset called $Q_2$, providing asymptotic estimates and analyzing families with limited size variation.

Contribution

It establishes new asymptotic bounds for the maximum size of $Q_2$-free families in the Boolean lattice and characterizes the size limit for families with at most three subset sizes.

Findings

Lower bound: approximately 2N

Upper bound: approximately 2.283N

Max size for families with ≤3 sizes: about 2.207N

Abstract

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N), $ where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.