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 ae<currency> y in P iff f(x) ae<currency> 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 Q (N) 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 Q (n) in Q (N) and for a multicolor version of a poset Ramsey number.