Joint work with Tanmay Inamdar.
Abstract. We investigate global structural properties of linear orders of a fixed infinite size. It is classical that the countable linear orders and the continuum-sized orders exhibit contrasting behaviours. Modern results show that strong extensions of the Baire Category Theorem induce order on the first uncountable cardinal yielding structural properties aligned with those of the countable linear orders, such as admitting a finite basis. We prove here that starting from the second uncountable cardinal, there is a large class of cardinals exhibiting chaotic behaviour similar to the continuum. At these levels, the class of special Aronszajn lines is either empty or it has arbitrarily long decreasing and increasing chains as well as large antichains, yielding that any basis must have maximal possible size.
These findings enable us to answer a question of Moore who asked whether it is possible to lift to the second uncountable cardinal the assertion that every Aronszajn line contains a Countryman line. We answer this question in the negative, in fact refuting this strong form of the Souslin Hypothesis at all successors of regular uncountable cardinals.
The root cause of these non-structure theorems is the existence of canonical transfinite trees satisfying anti-Ramsey partition relations. We construct these trees and the anti-Ramsey witnesses by a combination of walks on ordinals, club guessing, strong colourings of three different types, and a bit of finite combinatorics.
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A preliminary preprint was uploaded to the arXiv on 11/October/2024. Today, we post an updated version with quite a few improvements. They are:
1. In Theorem A, we waived the requirement that mu be non-ineffable.
2. In Theorem A, we also get that the class under discussion is not well-founded.
3. Added an analog of Theorem A for successors of singulars and for inaccessibles – Theorem D.
4. In Theorem D (now Theorem E), we reduced the “V=L” hypothesis to squares and waived the requirement that kappa be non-subtle.
5. In Lemma 2.26, we now get a line not containing a kappa-monotone line. This allows to meaningfully reject Countryman lines also at inaccessible levels.
6. An introduction was added to Subsection 3.2 to explain the need for the projection maps obtained there.
7. Most of the results that were using Avoiding levels now use Narrow levels instead, enabling conclusions in the context of stationary reflection.
8. New club-guessing results were added to Section 4.
9. Sections 8 and 9 were merged as we now get the strong conclusions of Section 9 also in the scenarios of Section 8. The outcome is Section 8 and the results there do not anymore require that the set of vanishing levels be large.
10. Section 10 (now Section 9) went through a complete revolution in order to satisfy the new needs arising from the previous sections.
It’s a long paper, so some reading advice for those interested:
1) The starting point of this paper was Moore’s Aronszajn lines and the club filter. This construction is lifted up beyond
2) The most demanding section (in the authors’ opinion) is Section 9 where
3) Section 1 is introductory.
4) Sections 2 to 4 have new results, but the most technical results in these three sections are Lemma 4.26 and Lemma 4.27.
5) Section 5, 6, 7 are self-contained modulo terminology from previous sections about walks on ordinals and
6) Sections 8 and 9 can be read taking the results from the previous sections as a black box.
7) Section 10 connects the dots together proving all but one of the main theorems.