  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.
