20 problems

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…

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

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…

Demailly's conjecture. For every integer $s\geqslant 1$,

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…

Chudnovsky's conjecture.

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

Tightness conjecture. There exists an $\alpha$-evenly distributed point set $X\subset\mathbb{R}^d$ such that

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 $Z\subset\mathbb{P}^n$ 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 $\alpha(J)$…

One-point deletion conjecture. There exist $p_i\in P$ and $q_i\in Q$, together with some $A\in O(2)$ and $\vec{t}\in\mathbb{R}^2$, such that

Let $P\subset\mathbb R^2$ be a finite point set of size $n$. Elekes's collinear-triples conjecture. If $P$ determines $\Omega(n^2)$ collinear triples, then at least ten points of…

Codegree decomposition conjecture. If

Degree-jumping conic containment conjecture. Then $Z$ is contained in a single conic.

High-dimensional balanced non-group-balanced configuration conjecture. If $n$ is sufficiently large, then there exists a discrete subset of $\mathbb{R}^n$ that is balanced but not…