12 problems

Politusi's conjecture. Geproci sets in $\mathbb{P}^3$ in LGP do not exist apart from sets of four points.

Let $Z\subset\mathbb P^3$ be a set of at least $5$ points in linear general position, meaning that no $3$ points of $Z$ are collinear and no $4$ points are coplanar. Projection con…

Converse conjecture. Then

Inclusion-property conjecture. $X$ is arithmetically Cohen–Macaulay (ACM).

Triple-line conjecture. If

Bárány's conjecture. As $n\to\infty$, $\deg X$ goes to infinity. This conjecture asks whether every sufficiently large finite planar point set contains a pair belonging to arbitrar…

Let $C>0$ and let ${\mathcal P}\subset\mathbb R^D$ be a finite point set that is proper $D$-dimensional up to $C$. For $r<D$, let $S^{(r)}\subset\mathbb R^D$ range over affine subs…

Let ${\mathcal P}\subset\mathbb R^2$ be any finite point set, let $3\le k\le|{\mathcal P}|$, and let $C>0$. An intersection is proper 2-dimensional up to $C$ in the sense used by t…

Let ${\mathcal P}\subset\mathbb R^2$ be a finite point set. An isometry is a distance-preserving map $\varphi:\mathbb R^2\to\mathbb R^2$. The isometry overlap conjecture. There is…

Let $S$ be a finite set of $n$ points in $\mathbb{R}^3$, let $m$ be the number of planes determined by $S$, and let $t$ be the number of lines determined by $S$. Assume that $n$ is…

Let a finite set of points lie in $\mathbb{R}^d$. A determined hyperplane is a hyperplane spanned by points of the set, and a $(d-2)$-flat is an affine subspace of dimension $d-2$…

Let $2n$ be an even number of noncollinear points in the plane. An ordinary line is a line containing exactly two of the points. Motzkin–Dirac's conjecture. Among any set of $2n$ n…