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

TL;DR

The paper proves that if an $eth_2$-Aronszajn line exists, then one can find such a line without an $eth_2$-Countryman subline, revealing new non-structure results for higher Aronszajn lines using advanced combinatorial methods.

Contribution

It introduces new non-structure theorems for higher Aronszajn lines and trees, contrasting with Moore's Basis Theorem, through novel combinatorial techniques involving walks and club guessing.

Findings

Existence of $eth_2$-Aronszajn lines without $eth_2$-Countryman lines.

Non-structure theorems for Aronszajn lines and trees at various cardinal successors.

Application of walks on ordinals and club guessing in higher set theory contexts.

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.