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)
.