Poset Ramsey number $R(P,Q_n)$. III. Chain Compositions and Antichains

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 $Q_n$.