Uniforming n-place Functions on Well Founded Trees

In this paper the Erdős-Rado theorem is generalized to the class of well founded trees. We define an equivalence relation on the class ds(∞)0 ( finite sequences of decreasing sequences of ordinals) with א0 equivalence classes, and for n < ω a notion of n-end-uniformity for a colouring of ds(∞)0 with μ colours. We then show that for every ordinal α, n < ω and cardinal μ there is an ordinal λ so that for any colouring c of T = ds(λ)0 with μ colours, T contains S isomorphic to ds(α) so that c↾S0 is n-end uniform. For c with domain T this is equivalent to finding S ⊆ T isomorphic to ds(α) so that c↾S depends only on the equivalence class of the defined relation, so in particular T → (ds(α)) μ,א0 . We also draw a conclusion on colourings of n-tuples from a scattered linear order.