Poset Ramsey Numbers for Boolean Lattices

A subposet Q' of a poset Q is a copy of a poset P if there is a bijection f between elements of P and Q' such that x ≤ y in P iff f(x) ≤ f(y) in Q'. For posets P, P', let the 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 Order (2017), where they proved R(Q₂, Q_n) ≤ 2n + 2 and R(Q_n, Q_m) ≤ mn + n + m, where Q_n is the Boolean lattice of dimension n. 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 ∈ N. In particular, we prove that R(Q_n, Q_n) ≤ n² - n + 2, R(Q₂, Q_n) ≤ (5/3)n + 2, and R(Q₃, Q_n) ≤ (37/16)n + 39/16. We also prove that R(Q₂,Q₃) = 5, and R(Q_m, Q_n) ≤ (m - 2 + (9m - 9)((2m - 3)(m + 1)))n + m + 3 for all n ≥ m ≥ 4.