Poset Ramsey Numbers for Boolean Lattices

For each positive integer n, let Q(n) denote the Boolean lattice of dimension n. For posets P, P', define the poset Ramsey number R(P, P') 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' with all elements blue. Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q(2), Q(n)) <= 2n + 2 and R(Q(n), Q(m)) <= mn + n + m. They later proved 2n <= R(Q(n), Q(n)) <= n(2) + 2n. Walzer later proved R(Q(n), Q(n)) <= n(2)+1. We provide some improved bounds for R(Q(n), Q(m)) for various n, m is an element of N. In particular, we prove that R(Q(n), Q(n)) <= n(2) - n + 2, R(Q(2), Q(n)) <= 5/3 n + 2, and R(Q(3), Q(n)) <= inverted right perpendicular37/16n + 55/16inverted left perpendicular. We also prove that R(Q(2), Q(3)) = 5, and R(Q(m), Q(n)) <= inverted right perpendicular(m - 1 + 2/m+1) n + 1/3 m + 2inverted left perpendicular for all n > m >= 4.