Every (𝜆⁺,κ⁺)-regular ultrafilter is (𝜆,κ)-regular

Source abstract

We prove the following: Theorem. If D D is a ( λ + , ϰ ) (\lambda ^+,\varkappa ) -regular ultrafilter, then either [(a)] D D is ( λ , ϰ ) (\lambda ,\varkappa ) -regular, or [(b)] the cofinality of the linear order ∏ D ⟨ λ , > ⟩ \prod _D\langle \lambda ,>\rangle is cf ⁡ ϰ \operatorname {cf}\varkappa , and D D is ( λ , ϰ ′ ) (\lambda ,\varkappa ’) -regular for all ϰ ′ > ϰ \varkappa ’>\varkappa . Corollary. Suppose that ϰ \varkappa is singular, ϰ > λ \varkappa >\lambda and either λ \lambda is regular, or cf ⁡ ϰ > cf ⁡ λ \operatorname {cf}\varkappa >\operatorname {cf}\lambda . Then every ( λ + n , ϰ ) (\lambda ^{+n},\varkappa ) -regular ultrafilter is ( λ , ϰ ) (\lambda ,\varkappa ) -regular. We also discuss some consequences and variations.