Poset Ramsey number R(P,Q). III. Chain compositions and antichains

An induced subposet (P2,≤2) of a poset (P1,≤1) is a subset of P1 such that for every two X,Y∈P2, X≤2Y if and only if X≤1Y. The Boolean lattice Qn of dimension n is the poset consisting of all subsets of {1,…,n} ordered by inclusion. Given two posets P1 and P2 the poset Ramsey number R(P1,P2) is the smallest integer N such that in any blue/red coloring of the elements of QN there is either a monochromatically blue induced subposet isomorphic to P1 or a monochromatically red induced subposet isomorphic to P2. We provide upper bounds on R(P,Qn) 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 Q2, 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,Qn) for posets P with at most 4 elements. If P is an antichain At on t elements, we show that R(At,Qn)=n+3 for 3≤t≤log⁡log⁡n. Additionally, we briefly survey proof techniques in the poset Ramsey setting P versus Qn.