An infinite combinatorial statement with a poset parameter

Abstract

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]ⁿ→[κ]^<λ 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 $(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1} {2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor }$ , for every infinite cardinal λ and all positive integers n and r with 2≤r<n. For example, (ℵ₂₁₀,4,ℵ₀)→32,768. Other order-dimension estimates yield relations such as (ℵ₁₀₉,4,ℵ₀) → 257 (using an estimate by Füredi and Kahn) and (ℵ₇,4,ℵ₀)→10 (using an exact estimate by Dushnik).