### Uniforming $n$-place functions on well founded trees

by Gruenhut and Shelah. [GhSh:909]

Set Theory and Its Applications, 2011

In this paper the Erdos-Rado theorem is generalized to the
class of well founded trees. We define an equivalence relation
on
the class rs (infty)^{< aleph_0} (finite sequences of
decreasing sequences of ordinals) with aleph_0 equivalence
classes, and for n< omega a notion of n-end-uniformity for a
colouring of rs (infty)^{< aleph_0} with mu colours. We
then show that for every ordinal alpha, n< omega and cardinal
mu there is an ordinal lambda so that for any colouring c ofT= rs (lambda)^{< aleph_0} with mu colours, T contains
S isomorphic to rs (alpha) so that c restriction
S^{< aleph_0}
is n-end uniform. For c with domain T^n this is equivalent to

finding S subseteq T isomorphic to rs (alpha) so that
c upharpoonright S^{n} depends only on the equivalence class
of
the defined relation, so in particular T-> (rs (alpha))^n_{mu,
aleph_0} . We also draw a conclusion on
colourings of n-tuples from a scattered linear order.

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