Boolean Lattices: Ramsey Properties and Embeddings

A subposet Q′ of a poset Q is a copy of a poset P if there is a bijection f between elements of P and Q′ such that x ≤ y in P iff f(x) ≤ f(y) in Q′. For posets P, P ′, let the poset Ramsey number R(P, P ′) be the smallest N such that no matter how the elements of the Boolean lattice QN are colored red and blue, there is a copy of P with all red elements or a copy of P ′ with all blue elements. We provide some general bounds on R(P, P ′) and focus on the situation when P and P ′ are both Boolean lattices. In addition, we give asymptotically tight bounds for the number of copies of Qn in QN and for a multicolor version of a poset Ramsey number.