Poset Ramsey Numbers for Boolean Lattices

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