A Decomposition Theorem for Aronszajn Lines

Keegan Dasilva Barbosa

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arXiv · 2003.12115

A Decomposition Theorem for Aronszajn Lines

Keegan Dasilva Barbosa

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Abstract

We show that under the proper forcing axiom the class of all Aronszajn lines behave like $\sigma$-scattered orders under the embeddability relation. In particular, we are able to show that the class of better quasi order labeled fragmented Aronszajn lines is itself a better quasi order. Moreover, we show that every better quasi order labeled Aronszajn line can be expressed as a finite sum of labeled types which are algebraically indecomposable. By encoding lines with finite labeled trees, we are also able to deduce a decomposition result, that for every Aronszajn line $L$ there is integer n such that for any finite colouring of $L$ there is subset $L^\prime$ of $L$ isomorphic to $L$ which uses no more than n colours.

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Keegan Dasilva Barbosa. 2020-03-26. A Decomposition Theorem for Aronszajn Lines. https://arxiv.org/abs/2003.12115

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