For every infinite cardinal k, the class of linear orders of size k has 2^k many distinct types, but this does not mean that this class is totally chaotic. For instance, it is conceivable that this class admits a small basis (collection of minimal elements) or a small universal family (collection of maximal elements). Indeed, as a prototype example, the class of countable linear orders has a 2-element basis and a 1-element universal family.

In this talk, we shall discuss some of the set theory generated by the attempts to tame classes of infinite linear orders, starting with the works of Cantor, Sierpinski, Souslin and Specker, and going through modern works.

Our contribution settles a higher analog of a conjecture of Shelah proposed by Justin Moore.

This is joint work with Tanmay Inamdar.