Partition theorems for expanded trees

We look for partition theorems for large subtrees for suitable uncountable trees and colourings. We concentrate on sub-trees of $^{κ\ge} 2$ 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.

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