6 problems

For a finite set of $n$ points in $\mathbb{E}^d$, its diameter graph is the graph whose vertices are the points and whose edges join pairs at the diameter of the set. A clique of s…

Let $e(d,n)$ denote the maximum number of diameters among sets of $n$ points in $\mathbb{R}^d$. For $d=3$, this is the Vázsonyi problem. Vázsonyi's conjecture. Vázsonyi conjectured…

Let $S_1$ and $S_2$ be two unit simplices in $\mathbb R^d$ forming a set of diameter $1$, with $k$ and $m$ vertices respectively, and assume $k\geq m$. Schur's conjecture. The simp…

Let an almost-equidistant diameter set be a set of points in $\mathbb R^d$ of diameter $1$ in which every three points contain a pair at distance $1$. The extremal bound conjecture…

Let $\mathrm{DIAM}(d,k,n)$ be the maximum number of unit $k$-simplices spanned by an $n$-point set in $\mathbb{R}^d$ of diameter $1$. Conjecture on unit simplices in diameter graph…

A unit regular simplex on $d$ vertices in $\mathbb R^d$ is a regular simplex whose edges have length $1$. Let two such simplices be given. Morić–Pach's conjecture. Any two unit reg…