Poset Ramsey Numbers for Boolean Lattices

Abstract

For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets P, P′, define the poset Ramsey numberR(P, P′) to be the least N such that for any red/blue coloring of the elements of QN, there exists either a subposet isomorphic to P with all elements red, or a subposet isomorphic to P′ with all elements blue. Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q2,Qn) ≤ 2n + 2 and R(Qn,Qm) ≤ mn + n + m. They later proved 2n ≤ R(Qn,Qn) ≤ n2 + 2n. Walzer later proved R(Qn,Qn) ≤ n2 + 1. We provide some improved bounds for R(Qn,Qm) for various n, m∈ ℕ. In particular, we prove that R(Qn,Qn) ≤ n2 − n + 2, R(Q2,Qn)≤53n+2, and R(Q3,Qn)≤⌈3716n+5516⌉. We also prove that R(Q2,Q3) = 5, and R(Qm,Qn)≤⌈(m−1+2m+1)n+13m+2⌉ for all n > m ≥ 4.