Mathematical Research Letters

Volume 25 (2018)

Number 6

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

Pages: 1957 – 1975



Hossein Lamei Ramandi (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.)

Justin Tatch Moore (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.)


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.

Received 20 April 2016

Accepted 4 July 2017

Published 25 March 2019