Poset Ramsey Numbers for Boolean Lattices

Abstract

A subposet $Q'$ of a poset $Q$ is a \textit{copy of a poset} $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x \le y$ in $P$ iff $f(x) \le f(y)$ in $Q'$. For posets $P, P'$, let the \textit{poset Ramsey number} $R(P,P')$ be the smallest $N$ such that no matter how the elements of the Boolean lattice $Q_N$ are colored red and blue, there is a copy of $P$ with all red elements or a copy of $P'$ with all blue elements. Axenovich and Walzer introduced this concept in \textit{Order} (2017), where they proved $R(Q_2, Q_n) \le 2n + 2$ and $R(Q_n, Q_m) \le mn + n + m$, where $Q_n$ is the Boolean lattice of dimension $n$. They later proved $2n \le R(Q_n, Q_n) \le n^2 + 2n$. Walzer later proved $R(Q_n, Q_n) \le n^2 + 1$. We provide some improved bounds for $R(Q_n, Q_m)$ for various $n,m \in \mathbb{N}$. In particular, we prove that $R(Q_n, Q_n) \le n^2 - n + 2$, $R(Q_2, Q_n) \le \frac{5}{3}n + 2$, and $R(Q_3, Q_n) \le \frac{37}{16}n + \frac{39}{16}$. We also prove that $R(Q_2,Q_3) = 5$, and $R(Q_m, Q_n) \le (m - 2 + \frac{9m - 9}{(2m - 3)(m + 1)})n + m + 3$ for all $n \ge m \ge 4$.