On minimal non-$\sigma$-scattered linear orders

Abstract

The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-$\sigma$-scattered. Specifically, we will show that Jensen's principle $\diamondsuit$ implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-$\sigma$-scattered linear orders of cardinality greater than $\aleph_1$, as given a successor cardinal $\kappa^+$, we obtain such linear orderings of cardinality $\kappa^+$ with the additional property that their square is the union of $\kappa$-many chains. We give two constructions: directly building such examples using forcing, and also deriving their existence from combinatorial principles. The latter approach shows that such minimal non-$\sigma$-scattered linear orders of cardinality $\kappa^+$ exist for every cardinal $\kappa$ in G\"odel's constructible universe, and also (using work of Rinot) that examples must exist at successors of singular strong limit cardinals in the absence of inner models satisfying the existence of a measurable cardinal $\mu$ of Mitchell order $\mu^{++}$.