14 problems

Let $X$ be an $n$-dimensional normed space, and let its equilateral dimension be the maximal cardinality of a subset of $X$ whose distinct points all have the same pairwise distanc…

Kusner's conjecture.

Füredi–Lagarias–Morgan conjecture. There exists an $\epsilon>0$ such that every equilateral set in $M$ satisfies

Let $\mathbb{M}_Z^n$ be the $n$-dimensional Minkowski space whose unit ball is $\operatorname{conv}([0,1]^n\cup[-1,0]^n)$. An equilateral set is a set of points whose pairwise dist…

Equilateral number conjecture. Every $n$-dimensional normed space $X$ satisfies

Let $X^3$ be a three-dimensional normed space with a differentiable norm, and let $e(X^3)$ denote the largest possible size of an equilateral set in $X^3$. Lawlor–Morgan's conjectu…

Let $X$ be an $n$-dimensional normed space, and let $e(X)$ denote the maximum cardinality of an equilateral subset of $X$. Lower-bound conjecture. If $\dim X=n$, then … This is the…

Let $d4a3^n$ be equipped with the tropical metric, and let $\mathrm{e}_{\mathrm{tr}}(\mathbb{R}^{n})$ denote the maximal cardinality of a tropical equilateral subset of $d4a3^n$. T…

Let $e_{\|.\|}$ be the largest number of vertices in an equilateral set in the normed space $(\mathbb{R}^d,\|.\|)$, and let $\ell_1^d$ denote $\mathbb{R}^d$ equipped with the…

Thompson's conjecture. For every $k\in\mathbb{N}\setminus\{0,1\}$, the value $d(k)$ can be $k-1$.

Let $X$ be a normed vector space of dimension $n$. An equilateral set is a set of points whose pairwise distances are all equal. Equilateral-set conjecture. Every normed $n$-dimens…

PFA conjecture. Assuming the Proper Forcing Axiom, every non-metrisable, perfectly normal compact space $K$ contains a non-metrisable, totally disconnected closed subspace.

Let $X$ be an $n$-dimensional normed space. The equilateral dimension conjecture. … Here $e(X)$ denotes the maximal cardinality of an equilateral set in $X$, that is, a set of pair…

Let $X$ be a finite-dimensional normed space, and let $m(X)$ denote the smallest possible cardinality of an equilateral subset of $X$ that is maximal with respect to inclusion. Swa…