Poset Ramsey Number $$R(P,Q_{n})$$. I. Complete Multipartite Posets

Abstract A poset $$(P^{\prime },\le _{P^{\prime }})$$ ( P ′ , ≤ ) contains a copy of some other $$(P,\le _{P})$$ if there is an injection $$f:P'\rightarrow P$$ f : → where for every $$X,Y\in X Y ∈ , $$X\le _{P} Y$$ and only $$f(X)\le _{P'} f(Y)$$ . For any posets P Q the Ramsey number R ( ) smallest integer N such that blue/red coloring Boolean lattice dimension either with all elements blue or red. complete $$\ell $$ ℓ -partite $$K_{t_{1},\dots ,t_{\ell }}$$ K t 1 ⋯ on $$\sum _{i=1}^{\ell } t_{i}$$ ∑ i = elements, which are partitioned into pairwise disjoint sets $$A^{i}$$ A $$|A^{i}|=t_{i}$$ | $$1\le i\le \ell two $$X\in A^{i}$$ $$Y\in A^{j}$$ j $$X<Y$$ < $$i<j$$ In this paper we show $$R(K_{t_{1},\dots }},Q_{n})\le n+\frac{(2+o_{n}(1))\ell n}{\log n}$$ R Q n + 2 o log