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Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, wesay that $P'$ contains a copy of $P$ if for some injective function $f\colonP\rightarrow P'$ and for any $A, B\in P$, $A\leq _P B$ if and only if$f(A)\leq_{P'} f(B)$. 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 elementsof an $N$-dimensional Boolean lattice are colored in blue and red, there iseither a copy of $P$ with all blue elements or a copy of $Q$ with all redelements.  We focus on the poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an$n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. It is known that$n+c_1(P) \leq R(P,Q_n) \leq c_2(P) n$, for positive constants $c_1$ and $c_2$.However, there is no poset $P$ known, for which $R(P, Q_n)> (1+\epsilon)n$, for$\epsilon >0$. This paper is devoted to a new method for finding upper boundson $R(P, Q_n)$ using a duality between copies of $Q_n$ and sets of elementsthat cover them, referred to as blockers. We prove several properties ofblockers and their direct relation to the Ramsey numbers. Using theseproperties we show that $R(\mathcal{N},Q_n)=n+\Theta(n/\log n)$, for a poset$\mathcal{N}$ with four elements $A, B, C, $ and $D$, such that $A<C$, $B<D$,$B<C$, and the remaining pairs of elements are incomparable.

Poset Ramsey number $R(P,Q_n)$. II. N-shaped poset