  Motivated by the paper of Axenovich and Walzer [2], we study the Ramsey-type
problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the
smallest Boolean lattice $\mathcal{B}_N$ such that any coloring on elements of
$\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$. This number
$N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper.
  Particularly, we determine the exact values of the Boolean rainbow Ramsey
number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains.
From these results, we also give some general upper and lower bounds of the
Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset
parameters.
