Partition theorems for expanded trees

TL;DR

This paper establishes partition theorems for large subtrees of uncountable trees with level-preserving embeddings, providing consistency results without large cardinals and aiming at applications in model theory.

Contribution

It introduces new partition theorems for expanded uncountable trees that preserve level equality rather than height, differing from prior work.

Findings

Proves partition theorems for uncountable trees with level-preserving embeddings

Achieves consistency results without large cardinal assumptions

Aims to apply results to model theoretic problems

Abstract

We look for partition theorems for large subtrees for suitable uncountable trees and colourings. We concentrate on sub-trees of $^{\kappa \ge} 2$ expanded by a well ordering of each level. Unlike earlier works, we do not ask the embedding to preserve the height of the tree but the equality of levels is preserved. We get consistency results without large cardinals. The intention is to apply it to model theoretic problems.