A Construction for Boolean Cube Ramsey Numbers

Let Q(n) be the poset that consists of all subsets of a fixed n-element set, ordered by set inclusion. The poset cube Ramsey number R(Q(n), Q(n)) is defined as the least m such that any 2-coloring of the elements of Q(m) admits a monochromatic copy of Q(n). The trivial lower bound R(Q(n), Q(n)) >= 2n was improved by Cox and Stolee, who showed R(Q(n), Q(n)) >= 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(Q(n), Q(n)) >= 2n + 1 for all n >= 3. The best known upper bound, due to Lu and Thompson, is R(Q(n), Q(n)) <= n(2) - 2n + 2.