Roy Shalev is finishing his doctorate working with Assaf Rinot at Bar Ilan University, and in this paper posted to ArXiv yesterday he answers a question left open by our work with Cummings and Moore on uncountable linear orderings. Our result shows that under the axiom V=L, for any successor cardinal \(\kappa^+\) there is a linear ordering of cardinality \(\kappa^+\) that is minimal with respect to being non-\(\sigma\)-scattered. We conjectured a similar result should hold in L at non-weakly compact inaccessible cardinals, and Shalev establishes this (and much more) using the Brodsky-Rinot proxy principle. The abstract of the paper is below, and it can be found here.
[Shalev]
Assuming an instance of the Brodsky-Rinot proxy principle holding at a regular uncountable cardinal κ, we construct \(2^\kappa\)-many pairwise non-embeddable minimal non-σ-scattered linear orders of size κ. In particular, in Gödel's constructible universe L, these linear orders exist for any regular uncountable cardinal κ that is not weakly compact. This extends a recent result of Cummings, Eisworth and Moore that takes care of all the successor cardinals of L. At the level of \(\aleph_1\), their work answered an old question of Baumgartner by constructing from ♢ a minimal Aronszajn line that is not Souslin. Our use of the proxy principle yields the same conclusion from a weaker assumption which holds for instance in the generic extension after adding a single Cohen real to a model of CH.