The four-point obstruction conjecture for infinite ultrametric spaces

Let $(X,d)$ be an infinite ultrametric space. A four-point subspace means a subspace of $(X,d)$ whose underlying set has four elements; $(X,d)$ belongs to ${\bf US}$ when it is an ultrametric space generated by a labeled star graph. The spaces $(X_4,d_4)$ and $(Y_4,\rho_4)$ are the four-point ultrametric spaces specified earlier in the paper, and “weakly similar” has the paper's stated meaning.

Four-point obstruction conjecture. The following statements are equivalent:

1. $(X,d) \notin {\bf US}$.
2. $(X,d)$ contains a four-point subspace which is weakly similar either to $(X_4,d_4)$ or to $(Y_4,\rho_4)$.

This conjecture extends the corresponding finite or limit-point characterization to all infinite ultrametric spaces; the paper presents it as the original problem motivating the work, so its resolution remains open here.