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We investigate a combinatorial game on ${\mathrm{\omega <\#comment/>}}_1$and show that mild large cardinal assumptions imply that every normal ideal on ${\mathrm{\omega <\#comment/>}}_1$satisfies a weak version of precipitousness. As an application, we show that the Raghavan-Todorčević proof of a longstanding conjecture of Galvin (done assuming the existence of a Woodin cardinal) can be pushed through under much weaker large cardinal assumptions [Forum Math. Pi 8 (2020), p. e15]. 

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 diamond principle 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 aleph1, 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 infinite cardinal kappa in Gödel's constructible universe, and also (using work of Rinot [28]) 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++.