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 $X,Y∈P, X≤_PY$ 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 $K_{t1,…,tℓ}$ is a poset on $∑^ℓ_{i=1}t_i$ elements, which are partitioned into $ℓ$ pairwise disjoint sets $A^i$ with $|A^i|=t_i, 1≤i≤ℓ$, such that for any two $X∈A^i$ and $Y∈A^j$, $X<Y$ if and only if $i<j$. In this paper we show that $R(K_{t1,…,tℓ, }Q_n)≤n+\frac{(2+on(1))ℓn}{logn}$.