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 Qₙ 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 QN 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(Qₘ,Qₙ)≤ nm-(1-o(1))nlog m where n≥ m and m is sufficiently large. This improves the best known upper bound on R(Qₙ,Qₙ) from n²-n+2 to n²-(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 Rʷ(P,Q) is the smallest N for which any blue/red-coloring of QN contains a blue weak copy of P or a red weak copy of Q. We show that Rʷ(Qₙ,Qₙ)≤ 0.96n².