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.
Categories: math.CO
