  An induced subposet $(P_2,\le_2)$ of a poset $(P_1,\le_1)$ is a subset of
$P_1$ such that for every two $X,Y\in P_2$, $X\le_2 Y$ if and only if $X\le_1
Y$. The Boolean lattice $Q_n$ of dimension $n$ is the poset consisting of all
subsets of $\{1,\dots,n\}$ ordered by inclusion. Given two posets $P_1$ and
$P_2$ the poset Ramsey number $R(P_1,P_2)$ is the smallest integer $N$ such
that in any blue/red coloring of the elements of $Q_N$ there is either a
monochromatically blue induced subposet isomorphic to $P_1$ or a
monochromatically red induced subposet isomorphic to $P_2$.
  We provide upper bounds on $R(P,Q_n)$ for two classes of $P$: parallel
compositions of chains, i.e.\ posets consisting of disjoint chains which are
pairwise element-wise incomparable, as well as subdivided $Q_2$, which are
posets obtained from two parallel chains by adding a common minimal and a
common maximal element. This completes the determination of $R(P,Q_n)$ for
posets $P$ with at most $4$ elements. If $P$ is an antichain $A_t$ on $t$
elements, we show that $R(A_t,Q_n)=n+3$ for $3\le t\le \log \log n$.
Additionally, we briefly survey proof techniques in the poset Ramsey setting
$P$ versus $Q_n$.
