Diagonal poset Ramsey numbers

A poset (Q,≤_Q) contains an induced copy of a poset (P,≤_P) if there exists an injective mapping ϕ P→ Q such that for any two elements X,Y∈ P, X≤_P Y if and only if ϕ(X)≤_Q ϕ(Y). By Q_n we denote the Boolean lattice (2^[n],⊆). The poset Ramsey number R(P,Q) for posets P and Q is the least integer N for which any coloring of the elements of Q_N in blue and red contains either a blue induced copy of P or a red induced copy of Q. In this paper, we show that R(Q_m,Q_n)≤ nm-(1-o(1))nlog m where n≥ m and m is sufficiently large. This improves the best known upper bound on R(Q_n,Q_n) from n^2-n+2 to n^2-(1-o(1)) nlog n. Furthermore, we determine R(P,P) where P is an n-fork or n-diamond up to an additive constant of 2. A poset (Q,≤_Q) contains a weak copy of (P,≤_P) if there is an injection ψ P→ Q such that ψ(X)≤_Q ψ(Y) for any X,Y∈ P with X≤_P Y. The weak poset Ramsey number R^w(P,Q) is the smallest N for which any blue/red-coloring of Q_N contains a blue weak copy of P or a red weak copy of Q. We show that R^w(Q_n,Q_n)≤ 0.96n^2.