Uniform n-place functions on T\subseteq ds(\alpha)

TL;DR

This paper generalizes the Erdos-Rado theorem to well-founded trees, establishing conditions for uniform colorings of finite sequences within these trees, with implications for scattered linear orders.

Contribution

It introduces a new framework for uniform colorings on well-founded trees, extending classical combinatorial theorems to a broader class of structures.

Findings

Generalized Erdos-Rado theorem to well-founded trees

Established existence of large homogeneous subtrees for colorings

Connected results to colorings of n-tuples in scattered linear orders

Abstract

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 of T=rs(lambda)^{< aleph_0} with mu colours, T contains S isomorphic to rs(alpha) so that c rest 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.