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

An induced subposet ( P 2 , <= 2 ) of a poset ( P 1 , <= 1 ) is a subset of P 1 such that for every two X , Y E P 2 , X <= 2 Y if and only if X <= 1 Y . The Boolean lattice Q n of dimension n is the poset consisting of all subsets of { 1 , ..., 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 <= t <= log log n . Additionally, we briefly survey proof techniques in the poset Ramsey setting P versus Q n . (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).