A POLARIZED PARTITION RELATION FOR WEAKLY COMPACT CARDINALS USING ELEMENTARY SUBSTRUCTURES. - Jones, Albin L. (Journal of Symbolic Logic, 2006)
We show that if κ isa weakly compact cardinal, then (Multiple line equation(s) cannot be represented in ASCII text) for any ordinals α κ κand μ < κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partition κ X κ = ∪ K U ∪ L of κ x κ into m + μ pieces either there are A ϵ [κ], B ϵ [κ], and i < m with A x B ⊆ K or there are C ϵ [κ], D ϵ [κ], and j < μ with C x D ⊆ L. 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.