Decomposing Aronszajn lines

Journal of Mathematical Logic 23 (1) (2022)
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Abstract

We show that under the proper forcing axiom the class of all Aronszajn lines behave like [Formula: see text]-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 [Formula: see text] there is an integer [Formula: see text] such that for any finite coloring of [Formula: see text] there is subset [Formula: see text] of [Formula: see text] isomorphic to [Formula: see text] which uses no more than [Formula: see text] colors.

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Introduction to Set Theory.K. Hrbacek & T. Jech - 2001 - Studia Logica 69 (3):448-449.

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