Jonsson algebras in successor cardinals

Abstract

We shall show here that in many successor cardinals λ, there is a Jonsson algebra (in other words Jn(λ), or λ is not a Jonsson cardinal). In connection with this we show that, e.g., for every ultrafilterD over ω, in (ω_ω, <)^ω/D there is no increasing sequence of length$\aleph _{(2^\aleph 0)^ + }$. On Jonsson algebras see e.g. [1]; for successor λ⁺ = 2^λ there is a Jonsson algebra, (λ)⇒Jn(λ⁺) (due to Chang, Erdös and Hajnal) and even in$2^{\aleph _\alpha } = \aleph _{(\alpha + n)}$ ([3]). We give here a method to prove, e.g., (λ_ω+1) when$2^{\aleph _\alpha } \leqq \aleph _{(\omega + 1)} and Jn(2^{\aleph _0 } ) when 2^{\aleph _0 } = \aleph _{\alpha + 1,} \alpha< \omega _1$; and similar results for higher cardinals.