  A poset $(P',\le_{P'})$ contains a copy of some other poset $(P,\le_P)$ if
there is an injection $f\colon P'\to P$ where for every $X,Y\in P$, $X\le_P Y$
if and only if $f(X)\le_{P'} f(Y)$. For any posets $P$ and $Q$, the poset
Ramsey number $R(P,Q)$ is the smallest integer $N$ such that any blue/red
coloring of a Boolean lattice of dimension $N$ contains either a copy of $P$
with all elements blue or a copy of $Q$ with all elements red. We denote by
$K_{t_1,\dots,t_\ell}$ a complete $\ell$-partite poset, i.e.\ a poset
consisting of $\ell$ pairwise disjoint sets $A^i$ of size $t_i$, $1\le i\le
\ell$, such that for any $i,j\in\{1,\dots,\ell\}$ and any two $X\in A^{i}$ and
$Y\in A^{j}$, $X<Y$ if and only if $i<j$. In this paper we show that
$R(K_{t_1,\dots,t_\ell},Q_n)\le n+\frac{(2+o_n(1))\ell n}{\log n}$.
