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

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.

This is a preliminary preprint. More results are upcoming. For updates, please visit http://assafrinot.com/paper/71