Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
Logic
2024-10-14 v1
Abstract
It is proved that if there is an -Aronszajn line, then there is one that does not contain an -Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of size . 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.
Cite
@article{arxiv.2410.08757,
title = {Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines},
author = {Tanmay Inamdar and Assaf Rinot},
journal= {arXiv preprint arXiv:2410.08757},
year = {2024}
}
Comments
This is a preliminary preprint. More results are upcoming. For updates, please visit http://assafrinot.com/paper/71