  Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we
say that $P'$ contains a copy of $P$ if for some injective function $f\colon
P\rightarrow P'$ 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 elements
of an $N$-dimensional Boolean lattice are colored 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 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 bounds
on $R(P, Q_n)$ using a duality between copies of $Q_n$ and sets of elements
that cover them, referred to as blockers. We prove several properties of
blockers and their direct relation to the Ramsey numbers. Using these
properties 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.
