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))︁n log 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 by Lu and Thompson (2022) to n$^2$ − (︁1 − o(1))︁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 ,≤$_Q$ ) 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$^2$.