35 problems

The paper considers hyperspaces obtained from compact ultrametric spaces and compares them with the corresponding hyperspaces of compact real trees. Analogy conjecture. The analogy…

Let $G$ be a connected graph. A labeling $l:V(G)\to\mathbb{R}^+$ is non-degenerate in the sense used by the source, and $(V(G),d_l)$ denotes the associated pseudoultrametric space.…

Let $G$ be a connected graph, and let a labeling $l:V(G)\to\mathbb{R}^+$ be non-degenerate when the labeling satisfies the paper's non-degeneracy condition. Discreteness-local fini…

Let $i=1,2$ and let $T_i=T_i(l_i)$ be finite labeled trees with non-degenerate labelings $l_i:V(T_i)\to\mathbb{R}^+$. Suppose that $(V(T_1),d_{l_1})$ and $(V(T_2),d_{l_2})$ are…

The realization conjecture. There exists a finite ultrametric space $(X,d)$ such that

The isometry characterization. The equality

The weak-similarity characterization. The equality

Let $n\geq1$ be an integer, let $\log_2(n)$ denote the binary logarithm of $n$, and let $\lfloor\log_2(n)\rfloor$ be its integer part. For an ultrametric space $(X,d)$, let $C(X)$…

Let $(Y,\rho)$ be an ultrametric space with $|Y|\geq3$. A centered sphere is a sphere with a center in the underlying space. The three-point centered-sphere conjecture. The followi…

Let $(X,d)$) be an ultrametric space. For each $p\to X$, define … Let $\overline{\bf B}_X$ denote the set of all closed balls of $(X,d)$, and let $\bf Cs_X$ denote the family of ce…

Conjecture 2.8. If

$K_{1,3}$ characterization conjecture. The following statements are equivalent:

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

Let $(X,d)$ and $(Y,\rho)$ be ultrametric spaces, with completions $(\tilde{X},\tilde{d})$ and $(\tilde{Y},\tilde{\rho})$. Let $\bar{\mathbf{B}}_X^0$ and $\bar{\mathbf{B}}_Y^0$ den…

Let $(X,d)$ be an ultrametric space, and let $(\tilde{X},\tilde{d})$ be its completion. Write $\bar{\mathbf{B}}_X$ and $\bar{\mathbf{B}}_{\tilde{X}}$ for the balleans of closed bal…

Let $(X,d)$ be an infinite ultrametric space. A labeled ray is a ray equipped with a labeling that generates the indicated ultrametric space. A tree is starlike if it has exactly o…

Let $(X,d)$ be an infinite ultrametric space. A metric space is weakly similar to another when there is a similarity between them, allowing a positive rescaling of distances. Infin…

Let $(X,d)$ be an infinite ${\bf US}$-space, meaning an ultrametric space generated by a labeled star graph with a center $c$. Put … and let $d_0$ be the restriction of $d$ to…

Let $(X^*,d^*)$ be a finite ultrametric space. The space $X^*$ is a US-space when it belongs to the class ${\bf US}$ of ultrametric spaces generated by labeled star graphs. A four-…

Compactness characterization conjecture. The following statements are equivalent: $(X,d)$ is compact; and there is a labeled ray $R(l)$ generating $(X_0,d_0)$ such that

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

Right-ideal characterization conjecture. If $1_{\mathbf{P_{PU}}}$ belongs to $\mathbf{A}$, then

Let $T$ be a rayless countable tree, meaning that $T$ contains no ray. For a vertex labeling $l\colon V(T)\to\mathbb{R}^{+}$, let $d_l$ be the generated ultrametric. Compact-case a…

Let $T$ be an infinite, locally finite tree. For a vertex labeling $l\colon V(T)\to\mathbb{R}^{+}$, let $d_l$ be the generated ultrametric and let its completion be taken with resp…

Let $T$ be a tree and let $(u_n)_{n\in\mathbb{N}}$ be a sequence of distinct vertices of $T$. The convex hull of a subset of vertices is the smallest subtree containing it, and a t…