A polarized partition relation for weakly compact cardinals using elementary substructures

Abstract

We show that if κ is a weakly compact cardinal, then

(\vector{κ⁺}{κ}) → ((\vector{α}{κ})_m (\vector{κⁿ}{κ})_μ))^1,1

for any ordinals α < κ⁺ and μ < κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partition

κ × κ⁺ = ⋃_{i < m} K_i ∪ ⋃_{j < μ} L_j

of κ × κ⁺ into m + μ pieces either there are A ∈ [κ]^κ, B ∈ [κ⁺]^α, and i < m with A × B ⊆ K_i or there are C ∈ [κ]^κ, D ∈ [κ⁺]^κⁿ, and j < μ with C × D ⊆ L_j. Related results for measurable and almost measurable κ are also investigated. Our proofs of these relations involve the use of elementary substructures of set models of large fragments of ZFC.