There may be no minimal non-$\sigma$-scattered linear orders

In this paper we demonstrate that it is consistent, relative to the existence of a supercompact cardinal, that there is no linear order which is minimal with respect to being non-$\delta$-scattered. This shows that a theorem of Laver, which asserts that the class of $\delta$-scattered linear orders is well quasi-ordered, is sharp. We also prove that $\mathrm{PFA}^{+}$ implies that every non-$\delta$-scattered linear order either contains a real type, an Aronszajn type, or a ladder system indexed by a stationary subset of $\omega_1$, equipped with either the lexicographic or reverse lexicographic order. Our work immediately implies that $C$H is consistent with “no Aronszajn tree has a base of cardinality $\aleph_1$.” This gives an affirmative answer to a problem due to Baumgartner.

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Received 20 April 2016

Accepted 4 July 2017

Published 25 March 2019