On the continuous gradability of the cut-point orders of $\mathbb R$-trees

TL;DR

This paper investigates the properties of orders in $ ext{R}$-trees, showing that not all are continuously gradable and linking this to set theory, with several independence results based on axiomatic assumptions.

Contribution

It demonstrates that the class of branchwise-real tree orders is not fully characterized by continuous gradability and establishes a set-theoretic connection for these properties.

Findings

Not all branchwise-real tree orders are continuously gradable.

A branchwise-real tree order is continuously gradable iff every well-stratified subtree is $ ext{R}$-gradable.

Existence of non-continuously gradable trees depends on set-theoretic assumptions like $ ext{MA}_ ext{κ}$.

Abstract

An $\mathbb R$-tree is a certain kind of metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying $\mathbb R$-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part, I answer this question in the negative: there is a 'branchwise-real tree order' which is not 'continuously gradable'. In the second part, I show that a branchwise-real tree order is continuously gradable if and only if every well-stratified subtree is $\mathbb R$-gradable. This link with set theory is put to work in the third part answering refinements of the main question, yielding several independence results. For example, when $\kappa \geq \mathfrak c$, there is a branchwise-real tree order which is not continuously gradable, and which satisfies a property corresponding to $\kappa$-separability. Conversely, under Martin's Axiom at $\kappa$ such a tree does not exist.