Countable partition ordinals

The structure of ordinals of the form ωωβωωβ for countable ββ is studied. The main result is: Theorem 1. If  β<ω1β<ω1is the sum of one or two indecomposable ordinals, thenωωβ→(ωωβ,3)2.ωωβ→(ωωβ,3)2. Also an example is given to show that α→(α,3)2α→(α,3)2 need not imply α→(α,n)2α→(α,n)2 for all n<ωn<ω.