Two-cardinal versions of weak compactness: Partitions of triples

Abstract

Let κ be a regular uncountable cardinal, and λ be a cardinal greater than κ. Our main result asserts that if (λ^{< κ})^{<(λ< κ)} = λ^{< κ}, then (p_{κ, λ}(NIn_{κ, λ^{< κ}}))⁺ $\longrightarrow$ ((NS_{κ, λ}^{[λ]< κ})⁺, NS_{κ, λ}s⁺)³ and (p_{κ, λ}(NAIn_{κ, λ^{< κ}}))⁺ $\longrightarrow$ (NS_{κ, λ}s⁺)³, where NS_{κ, λ}s (respectively, NS_{κ, λ}^{[λ]< κ}) denotes the smallest seminormal (respectively, strongly normal) ideal on P_κ (λ), NIn_{κ, λ^{< κ}} (respectively, NAIn_{κ, λ^{< κ}}) denotes the ideal of non-ineffable (respectively, non-almost ineffable) subsets of P_κ(λ^{< κ}), and p_{κ, λ}: P_κ(λ^{< κ}) → P_κ (λ) is defined by p_{κ, λ}(x) = x ∩ λ.