Diagonal poset Ramsey numbers

A poset $(Q,\le_Q)$ contains an induced copy of a poset $(P,\le_P)$ if there exists an injective mapping $\phi\colon P\to Q$ such that for any two elements $X,Y\in P$, $X\le_P Y$ if and only if $\phi(X)\le_Q \phi(Y)$. By $Q_n$ we denote the Boolean lattice $(2^{[n]},\subseteq)$. 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 $R(Q_m,Q_n)\le nm-\big(1-o(1)\big)n\log m$ where $n\ge m$ and $m$ is sufficiently large. This improves the best known upper bound on $R(Q_n,Q_n)$ from $n^2-n+2$ to $n^2-\big(1-o(1)\big) n\log n$. Furthermore, we determine $R(P,P)$ where $P$ is an $n$-fork or $n$-diamond up to an additive constant of $2$. A poset $(Q,\le_Q)$ contains a weak copy of $(P,\le_P)$ if there is an injection $\psi\colon P\to Q$ such that $\psi(X)\le_Q \psi(Y)$ for any $X,Y\in P$ with $X\le_P Y$. The weak poset Ramsey number $R^{\text{w}}(P,Q)$ 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 $R^{\text{w}}(Q_n,Q_n)\le 0.96n^2$.