On minimal non-σ-scattered linear orderings

The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-σ-scattered. Specifically, we will show that Jensen's principle ♢ implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-σ-scattered linear orders of cardinality greater than ℵ ₁ , as given a successor cardinal κ+, we obtain such linear orderings of cardinality κ+ with the additional property that their square is the union of κ-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-σ-scattered linear orders of cardinality κ+ exist for every cardinal κ in Gödel'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 μ of Mitchell order μ++.