A Construction for Boolean Cube Ramsey Numbers

Let Qn be the poset that consists of all subsets of a fixed n-element set, ordered by set inclusion. The poset cube Ramsey number R(Qn,Qn) is defined as the least m such that any 2-coloring of the elements of Qm admits a monochromatic copy of Qn. The trivial lower bound R(Qn,Qn) ≥ 2n was improved by Cox and Stolee, who showed R(Qn,Qn) ≥ 2n + 1 for 3 ≤ n ≤ 8 and n ≥ 13 using a probabilistic existence proof. In this paper, we provide an explicit construction that establishes R(Qn,Qn) ≥ 2n + 1 for all n ≥ 3. The best known upper bound, due to Lu and Thompson, is R(Qn,Qn) ≤ n2 − 2n + 2.