Q-free Families in the Boolean Lattice.

For a family ${{\cal F}}$ of subsets of [ n] = {1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that ${{\cal F}}$ 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 be the poset with distinct elements a, b, c, d, a < b, c < d; i.e., the 2-dimensional Boolean lattice. We show that 2 N − o( N) ≤ ex( n, Q) ≤ 2.283261 N + o( N), where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest Q-free family of subsets of [ n] having at most three different sizes has at most 2.20711 N members. [ABSTRACT FROM AUTHOR]