Poset Ramsey Number R(P, Qn). I. Complete Multipartite Posets

A poset ( P ',<= P ') contains a copy of some other poset (P,<= P) if there is an injection f : P ' -> P where for every X, Y is an element of P, X <= P Y if and only if f ( X) = <=(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. A complete l-partite poset Kt(1),...,t(l) is a poset on Sigma(l)(i=1) t(i) elements, which are partitioned into l pairwise disjoint sets A(i) with |A(i)| = t(i), 1 <= i <= l , such that for any two X is an element of A(i) and Y is an element of A(j), X < Y if and only if i < j. In this paper we show that R(K-t1,..., t(l), Q(n)) <= n + (2+o(n)(1)) ln / log n.