Given partially ordered sets (posets) $(P, \leq _P\!)$ and $(P^{\prime}, \leq _{P^{\prime}}\!)$ , we say that $P^{\prime}$ contains a copy of $P$ if for some injective function $f\,:\, P\rightarrow P^{\prime}$ and for any $X, Y\in P$ , $X\leq _P Y$ if and only if $f(X)\leq _{P^{\prime}} f(Y)$ . For any posets $P$ and $Q$ , the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$ -dimensional Boolean lattice are coloured in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$ -dimensional Boolean lattice $Q_n$ , as $n$ grows large. We show a sharp jump in behaviour of this number as a function of $n$ depending on whether or not $P$ contains a copy of either a poset $V$ , that is a poset on elements $A, B, C$ such that $B\gt C$ , $A\gt C$ , and $A$ and $B$ incomparable, or a poset $\Lambda$ , its symmetric counterpart. Specifically, we prove that if $P$ contains a copy of $V$ or $\Lambda$ then $R(P, Q_n) \geq n +\frac{1}{15} \frac{n}{\log n}$ . Otherwise $R(P, Q_n) \leq n + c(P)$ for a constant $c(P)$ . This gives the first non-marginal improvement of a lower bound on poset Ramsey numbers and as a consequence gives $R(Q_2, Q_n) = n + \Theta \left(\frac{n}{\log n}\right)$ .
A 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 .
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