  Given posets $\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k$, let the {\em
Boolean Ramsey number} $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)$ be
the minimum number $n$ such that no matter how we color the elements in the
Boolean lattice $\mathbf{B}_n$ with $k$ colors, there always exists a poset
$\mathbf{P}_i$ contained in $\mathbf{B}_n$ whose elements are all colored with
$i$. This function was first introduced by Axenovich and Walzer~\cite{AW}.
Recently, many results on determining $R(\mathbf{B}_m,\mathbf{B}_n)$ have been
published. In this paper, we will study the function
$R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)$ for each $\mathbf{P}_i$'s
being the $V$-shaped poset. That is, a poset obtained by identifying the
minimal elements of two chains.
  Another major result presented in the paper is to determine the minimal
posets $\mathbf{Q}$ contained in $\mathbf{B}_n$, when
$R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)=n$ is determined, having the
Ramsey property described in the previous paragraph.
  In addition, we define the {\em Boolean rainbow Ramsey number}
$RR(\mathbf{P},\mathbf{Q})$ the minimum number $n$ such that when arbitrarily
coloring the elements in $\mathbf{B}_n$, there always exists either a
monochromatic $\mathbf{P}$ or a rainbow $\mathbf{Q}$ contained in
$\mathbf{B}_n$. The upper bound for $RR(\mathbf{P},\mathbf{A}_k)$ was given by
Chang, Li, Gerbner, Methuku, Nagy, Patkos, and Vizer for general poset
$\mathbf{P}$ and $k$-element antichain $\mathbf{A}_k$. We study the function
for $\mathbf{P}$ being the $V$-shaped posets in this paper as well.
