An infinite combinatorial statement with a poset parameter

We introduce an extension, indexed by a partially ordered set P and cardinal numbers κ, λ, denoted by (κ,<λ) ; P , of the classical relation (κ, n, λ) → ρ in infinite combinatorics. By definition, (κ, n, λ) → ρ holds if every map F : [κ] → [κ] has a ρ-element free set. For example, Kuratowski’s Free Set Theorem states that (κ, n, λ) → n + 1 holds iff κ ≥ λ+n, where λ+n denotes the n-th cardinal successor of an infinite cardinal λ. By using the (κ,<λ) ; P framework, we present a self-contained proof of the first author’s result that (λ+n, n, λ) → n + 2, for each infinite cardinal λ and each positive integer n, which solves a problem stated in the 1985 monograph of Erdős, Hajnal, Máté, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation (λ+(n−1) , r, λ) → 2 ⌊