$Q_2$-free families in the Boolean lattice

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 $2N-o(N)\le ex(n,Q_2)\le 2.283261N+o(N),$ where $N=\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$. 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.

18 pages, 2 figures