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

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.