A polarized partition relation for weakly compact cardinals using elementary substructures

Jones, Albin L.

J. Symbolic Logic, Tome 71 (2006) no. 1, p. 1342-1352 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2006-12-14
Classification:  elementary substructures, measurable cardinals, normal ultrafilters, polarized partition relations, Ramsey theory, transfinite numbers, weakly compact cardinals,  Primary 03E02, 05D10; Secondary 04A20