Jonsson algebras in successor cardinals

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 ultrafilter D over ω, in (ω_ω, <)^ω/D there is no increasing sequence of length {Mathematical expression}. 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 {Mathematical expression} ([3]). We give here a method to prove, e.g., (λ_ω+1) when {Mathematical expression}; and similar results for higher cardinals.