Decomposing Aronszajn lines

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.