Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines

Joint work with Tanmay Inamdar.

Abstract. It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of size $\aleph_1$.

The proof combines walks on ordinals, club guessing, strong colourings of three different types, and a bit of finite combinatorics.

This and further non-structure theorems for Aronszajn lines and trees are established for successors of regulars, successors of singulars, as well as inaccessibles.

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