I gave an invited talk at the Set Theory meeting in Obwerwolfach, January 2025.
Talk Title: Non-structure theorems for higher Aronszajn lines.
Abstract: An $\omega_1$-Countryman line is an uncountable linear order $L$ such that $L^2$ (with the pointwise ordering) is the union of countably many chains. In particular, any map from an uncountable subset of $L$ to $L$ must be monotone on some uncountable subset. Note that any $\omega_1$-Countryman line is an $\omega_1$-Aronszajn line, i.e., an uncountable linear order that is monotonically far from uncountable separable orders and from uncountable well-orders.
In a paper from 1976, Shelah proved that an $\omega_1$-Countryman line exists and he conjectured it is consistent that every $\omega_1$-Aronszajn line contains an $\omega_1$-Countryman line. In a paper from 2006, Moore verified Shelah’s conjecture, proving that the assertion follows from PFA. In a workshop at the American Institute of Mathematics in 2016, Moore asked whether it is likewise consistent that every $\omega_2$-Aronszajn line contains an $\omega_2$-Countryman line.
In this talk, we answer Moore’s question in the negative, proving (in ZFC) that for every regular uncountable cardinal $\lambda$, if there exists a $\lambda^+$-Aronszajn line, then there exists one without a $\lambda^+$-Countryman subline. Our results are not limited to successors of regulars. Indeed, from a mild assumption (that does not involve $\diamondsuit$ nor cardinal arithmetic, and holds in the constructible universe for every regular uncountable $\kappa$ that is not weakly compact), we obtain $2^\kappa$ many pairwise monotonically far $\kappa$-Aronszajn lines, each satisfying that for any of its subsets $L$ of size $\kappa$, there is a map from $L$ to $L$ that is not monotone on any $\kappa$-sized subset.
The proof combines walks on ordinals, club guessing, strong colorings of three different types, and a bit of finite combinatorics.
This is joint work with Tanmay Inamdar.
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