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

A poset $(P\prime ,{\le }_{P\prime })$ contains a copy of some other poset $(P,{\le }_P)$ if there is an injection $f:P\prime \to P$ where for every $X,Y\in P,X{\le }_{P}Y$ if and only if $f(X){\le }_{P\prime }f(Y)$. 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 $K_{t1,\dots ,t\ell }$ is a poset on ${\sum }_{i=1}^{\ell }{t}_i$ elements, which are partitioned into $\ell$ pairwise disjoint sets $A^i$ with $|A^i|=t_i,1\le i\le \ell$, such that for any two $X\in A^i$ and $Y\in A^j$, $X<Y$ if and only if $i<j$. In this paper we show that $R(K_{t1,\dots ,t\ell ,}{Q}_n)\le n+\frac{(2+on(1))\ell n}{logn}$.