Poset Ramsey numbers: large Boolean lattice versus a fixed poset

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 $f: P\rightarrow P'$ and for any $X, Y\in P$, $X\leq _P Y$ if and only of $f(X)\leq_{P'} f(Y)$. 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 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 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 $V$, i.e. a poset on elements $A, B, C$ such that $B>C$, $A>C$, 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 $R(Q_2, Q_n) = n + \Theta (\frac{n}{\log n})$.