Two-cardinal versions of weak compactness: Partitions of triples

Let k be a regular uncountable cardinal, and λ be a cardinal greater than k. Our main result asserts that if (λ<k)<(λ<k) = λ<k, then (pk,λ(NInk,λ<k))+ → + (NS[λ]<k k;λ )+; NSk;λs+)3 and (pk,λ(NAInk;λ<k))+ → (NSk;λs+)3, where NSk;λs (respectively, NS[λ]<k k;λ) denotes the smallest seminormal (respectively, strongly normal) ideal on Pk(λ), NInk,λ<k (respectively, NAInk;λ<k) denotes the ideal of non-ineffable (respectively, non-almost ineffable) subsets of Pk(λ<k), and pk,λ : Pk(λ<k) → Pk(λ) is defined by pk,λ(x) = x ∩ λ.