MFO workshop in Set Theory, January 2025

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 $\mathrm{♢}$ 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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