  The poset Ramsey number $R(Q_m,Q_n)$ is the smallest integer $N$ such that
any blue-red coloring of the elements of the Boolean lattice $Q_N$ has a blue
induced copy of $Q_m$ or a red induced copy of $Q_n$. The weak poset Ramsey
number $R_w(Q_m,Q_n)$ is defined analogously, with weak copies instead of
induced copies. It is easy to see that $R(Q_m,Q_n) \ge R_w(Q_m,Q_n)$.
  Axenovich and Walzer showed that $n+2 \le R(Q_2,Q_n) \le 2n+2$. Recently, Lu
and Thompson improved the upper bound to $\frac{5}{3}n+2$. In this paper, we
solve this problem asymptotically by showing that $R(Q_2,Q_n)=n+O(n/\log n)$.
  In the diagonal case, Cox and Stolee proved $R_w(Q_n,Q_n) \ge 2n+1$ using a
probabilistic construction. In the induced case, Bohman and Peng showed
$R(Q_n,Q_n) \ge 2n+1$ using an explicit construction. Improving these results,
we show that $R_w(Q_m,Q_n) \ge n+m+1$ for all $m \ge 2$ and large $n$ by giving
an explicit construction; in particular, we prove that $R_w(Q_2,Q_n)=n+3$.
