Boolean lattices: Ramsey properties and embeddings

الأصل · EN

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.