26 problems

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 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 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 $f: [0,\infty) \to [0,\infty)$ be a function. Let $\mathbf{A}$ be the class of all unbounded boundedly compact (proper) ultrametric spaces, and let $\mathbf{P}_{\mathbf{A}}$ de…

UGVL characterization conjecture. The following statements are equivalent:

Let $A$ and $B$ be nonempty spherically complete subsets of an ultrametric space, and let Statements (i)--(iv) denote the four statements from Theorem 1. Extension conjecture. Stat…

Fixed-point characterization. The following conditions are equivalent: (i) $F$ has a fixed point in every nonempty spherically complete subspace of $M$; and (ii) the displayed ineq…

Cauchy-hull conjecture. The following conditions are equivalent:

Metric characterization conjecture. The following statements are equivalent:

Let $(X,\rho)$ be an infinite Polish ultrametric space. A map $\varphi:X\rightarrow X$ is locally contractive if, for every $x\in X$, there is a neighborhood on which distances fro…

Let $X$ be a compact nonempty semimetric space and let $\varepsilon>0$. A compact semimetric approximation conjecture asserts that there exist continuous mappings … such that $Y$ i…