39 problems

Let ${\mathbb{X}}\subseteq {\mathbb{P}}^n$ be a set of points having $\mathop{\rm CBP}\nolimits(r)$. A plane configuration $\mathcal{P}$ is a union of distinct positive-dimensional…

Jamison's classification conjecture. Every odd-critical configuration with $n\geq 50$ points belongs to one of these four infinite families.

Witsenhausen's conjecture. The maximum is attained when the points are distributed among the $n+1$ vertices of a regular simplex of edge length $1$. Witsenhausen established the up…

Let $\mathcal{Q}_N$ denote the configuration space of $N$ points in three-dimensional space, and let $F_N$ be the angular sum defined in the paper. The asymptotic spherical-distrib…

Let $\mathcal{Q}_N$ denote the configuration space of $N$ points in $d$-dimensional space, and let $F_N$ be the angular sum defined in the paper. The eight-point maximum conjecture…

The five-point maximum conjecture.

Koszul point-configuration conjecture. If

Completeness conjecture. The list of twelve pairwise non-equivalent configurations of order five is complete.

Let $n$ blue points, no three on a line, and $n$ red points disjoint from the blue points be given. Assume that every line through two blue points contains a red point. Milićević's…

Let $P$ be a non-collinear set of $n$ two-colored points, and suppose that no line contains more than $(1/2-delta)n$ points, where $delta>0$. Böröczky extremality conjecture. For e…

The converse to the surplus condition criterion. One has

Global minimizer conjecture. The hexagonal lattice $\mathsf{A}_2$ is the global minimizer of these energies among point configurations at fixed density. The paper proves only local…

Let $6\mid n$ and let $E$ be any elliptic curve in $\mathbb{P}^2$. Apply the Elliptified Böröczky construction to the points of $E[n]$, and let $\mathcal{B}_{En}$ denote the triple…

Let a $d$-dimensional set of $n$ points in $obreakmathbb{R}^d$ be a set whose affine span has dimension $d$. Two lines have pairwise distinct directions when no two are parall…

Let $k$ be a positive integer, and let $G$ be a complete geometric graph on $2k$ points. A $k$-staircase is the point-set construction defined earlier in the source. Staircase char…

For a finite set of points in the plane, consider every line determined by at least two points of the set, and compare the numbers of points on its two sides. Stretchable-configura…

Let $S$ be a finite set of points in the plane, and consider lines containing at least two points of $S$. For each such line, count the points of $S$ on either side of it. Kupitz's…

Let a finite or periodic point set be given in the Euclidean plane $\mathbb R^2$, and let $\mathrm{PDD}$ denote its pointwise distance distribution invariant. PDD completeness conj…

Exact linear bound conjecture. The lower bound on $P(k)$ in Theorem is tight. Namely,

Central Delannoy formula. The number of subdivisions of a $2\times n$ grid is

Let $P$ be a set of $n$ points in general position in the plane, meaning that no three points of $P$ are collinear, and let $R$ be a set of $n$ points disjoint from $P$. Cubic-curv…

Let $m\geq 6$ and $c>0$. A set of points in the plane is said to determine a direction when one of the lines spanned by two of its points has that direction. Elekes's conjecture. T…

Let $P$ be a set of $n$ points in general position in the plane, where general position means that no three points are collinear. Let $c_0$ be an absolute constant and suppose that…

Urrutia's conjecture. The tight bound for the number of points that can always be guaranteed to be enclosed by every circle through some pair of points is $\frac{n}{4}\pm c$ for so…

Let $Z\subset\mathbb{P}^n$ be a set of $n+3$ points in linearly general position, with $n$ odd, and let $I_Z$ be its homogeneous ideal. For a homogeneous ideal $J$, let $\alpha(J)$…