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

Abstract

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