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.
تحميل البحث