The $k$-color poset Ramsey number of the diamond is $2k$

Abstract

The $k$-color Ramsey number of a poset $P$ is the smallest dimension $d$ such that any $k$-coloring of the $d$-dimensional Boolean lattice contains a monochromatic induced copy of $P$. For $ j\ge 2$ the $j$-diamond is the poset that consists of a maximum element, a minimum element and an antichain of size $j$. We prove that the $k$-color Ramsey number of the $j$-diamond is at most $jk$. Colorings of the Boolean lattice that assign sets of the same size the same color provide matching lower bounds for $j \in \{2,3,4\}$.