17 problems

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

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

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 $(X,\delta)$ be a finite ordinal space and let $n\in\mathbb N^+$. For a subset $A\subseteq X$, write $A\hookrightarrow\mathbb R^n$ when the induced ordinal space on $A$ embeds…

Let $(X,\delta)$ be an ordinal space with $|X|=n\in\mathbb N^+$, and let $x_1,x_2,\ldots,x_n$ be an enumeration of its points satisfying the majorization property. Write…

Low-dimensional special linear group conjecture. For every prime power $\mathfrak{q}$ there exist $D=D(\mathfrak{q})\geqslant1$ and positive constants $c=c(\mathfrak{q})$,…

Special linear group Euclidean distortion conjecture. For every $k\in\mathbb{N}$ and every prime power $\mathfrak{q}$,

Finite-dimensional snowflake embedding conjecture. For every $\upomega\in(0,\tfrac12)$, there exists $\mathsf{C}_\upomega>0$ such that, for every integer $k\geqslant 2$, the…

Let $(\mathcal{M},d_\mathcal{M})$ be an $n$-point metric space. For $p,D\geqslant 1$, a bi-Lipschitz distortion-$D$ embedding of $\mathcal{M}$ into a metric space means an embeddin…

Let $\mathcal{M}\subseteq \ell_1$ be a subset with $|\mathcal{M}|=n$. An embedding of $\mathcal{M}$ into a normed space has distortion $D$ if distances are preserved up to a multip…

Let $L$ be a finite set of graphs. A graph excludes $L$ as a minor if it contains no member of $L$ as a minor, where minors are obtained by edge contractions, edge deletions, and v…

A planar graph metric is a shortest-path metric on a finite planar graph whose edges have arbitrary weights. The Planar Conjecture. Every metric supported on a finite planar graph…

For $2<q<p$ and $m,n\in\mathbb N$, let $[m]_q^n$ denote the $n$-dimensional integer grid with its $\ell_q$ metric, and let $c_p([m]_q^n)$ be its least bi-Lipschitz distortion in…

$k$-sum embedding conjecture. For any family of graphs $\mathcal F$, we have

GNRS conjecture. For every family of finite graphs $\mathcal F$, one has $c_1(\mathcal F)=O(1)$ if and only if $\mathcal F$ forbids some minor.

Let $\mathbb H$ be the discrete Heisenberg group, let $B_n$ be its radius-$n$ ball with the word metric $d_W$, and let $c_1(B_n,d_W)$ denote the least distortion of an embedding of…

Gupta–Newman–Rabinovich–Sinclair conjecture. For every family of finite graphs $\mathcal F$, one has