  For two posets $(P,\le_P)$ and $(P',\le_{P'})$, we say that $P'$ contains a
copy of $P$ if there exists an injective function $f\colon P'\to P$ such that
for every two $X,Y\in P$, $X\le_P Y$ if and only if $f(X)\le_{P'} f(Y)$. Given
two posets $P$ and $Q$, let the poset Ramsey number $R(P,Q)$ be the smallest
integer $N$ such that any coloring of the elements of an $N$-dimensional
Boolean lattice in blue or red contains either a copy of $P$ where all elements
are blue or a copy of $Q$ where all elements are red. We determine the poset
Ramsey number $R(A_t,Q_n)$ of an antichain versus a Boolean lattice for small
$t$ by showing that $R(A_t,Q_n)=n+3$ for $3\le t\le \log \log n$.
