Eduardo Martinez Mendoza

arXiv2026

Countryman Lines and the Continuum Hypothesis

We explore the prospect of basis-like results for the class of Aronszajn lines which are consistent with the Continuum Hypothesis (CH). In particular, we prove that each of the following statements is consistent with CH: Any two Countryman lines contain isomorphic or anti-isomorphic uncountable suborders; for any coherent Aronszajan tree $T \subseteq {}^{< ω_1} ω$, the filter $\mathcal{U}(T)$ is an ultrafilter. The first result confirms a conjecture of Shelah in the context of CH which was previously shown to follow from the Proper Forcing Axiom. On the other hand, the weak diamond principle $2^ω< 2^{ω_1}$ implies that there does not exist a two element basis for the Countryman lines.

math.LO↗

arXiv2025

An almost Kurepa Suslin tree with strongly non-saturated square

For uncountable downwards closed subtrees $U$ and $W$ of an $ω_1$-tree $T$, we say that $U$ and $W$ are strongly almost disjoint if their intersection is a finite union of countable chains. The tree $T$ is strongly non-saturated if there exists a strongly almost disjoint family of $ω_2$-many uncountable downwards closed subtrees of $T$. In this article we construct a Knaster forcing which adds a Suslin tree together with a family of $ω_2$-many strongly almost disjoint automorphisms of it (and thus the square of the Suslin tree is strongly non-saturated). To achieve this goal, we introduce a new idea called $ρ$-separation, which is an adaptation to the finite context of the notion of separation which was recently introduced by Stejskalová and the first author for the purpose of adding automorphisms of a tree with a forcing with countable conditions.

math.LO↗