  We say that a poset $(Q,\le_{Q})$ contains an induced copy of a poset
$(P,\le_P)$, if there is an injective function $\phi\colon P\to Q$ such that
for every two $X,Y\in P$, $X\le_P Y$ if and only if $\phi(X)\le_Q \phi(Y)$.
Given a fixed $2$-coloring $c$ of a poset $P$, the poset Erd\H{o}s-Hajnal
number of this colored poset is the minimum integer $N$ such that every
$2$-coloring of the Boolean lattice $Q_N$ contains an induced copy of $P$
colored as in $c$, or a monochromatic induced copy of $Q_n$. We present bounds
on the poset Erd\H{o}s-Hajnal number of general colored posets, antichains,
chains, and small Boolean lattices.
  Let the poset Ramsey number $R(Q_n,Q_n)$ be the minimal $N$ such that every
$2$-coloring of $Q_N$ contains a monochromatic induced copy of $Q_n$. As a
corollary, we show that $R(Q_n,Q_n)> 2.24n$, improving on the best known lower
bound $2n+1$ by Cox and Stolee.
