Poset Ramsey Numbers for Boolean Lattices

Abstract

For each positive integer n, let Q_n denote the Boolean lattice of dimension n. For posets $P, P^{\prime }$, define the poset Ramsey number $R(P,P^{\prime })$ to be the least N such that for any red/blue coloring of the elements of Q_N, there exists either a subposet isomorphic to P with all elements red, or a subposet isomorphic to $P^{\prime }$ with all elements blue. Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q₂,Q_n) ≤ 2n + 2 and R(Q_n,Q_m) ≤ mn + n + m. They later proved 2n ≤ R(Q_n,Q_n) ≤ n² + 2n. Walzer later proved R(Q_n,Q_n) ≤ n² + 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) ≤ n² − n + 2, $R(Q_{2}, Q_{n}) \le \frac {5}{3}n + 2$, and $R(Q_{3}, Q_{n}) \le \lceil \frac {37}{16}n + \frac {55}{16}\rceil$. We also prove that R(Q₂,Q₃) = 5, and $R(Q_{m}, Q_{n}) \le \left \lceil \left (m - 1 + \frac {2}{m+1} \right )n + \frac {1}{3} m + 2\right \rceil$ for all n > m ≥ 4.