155 problems

For every $n\ge 2$ and complexity bound $E$, there is a constant $C_{n,E}$ such that the following holds. Let $0<\delta,\lambda\le 1$, let $\mathbb{T}$ be a family of unit-length…

Does there exist an absolute constant $c>0$ such that, for every finite set $P\subset\mathbb{R}^3$ with $|P|=N$, the number of distinct pairwise distances satisfies…

Let $\mathbb{F}$ be a field, let $0<\alpha\leq 1$, and let a line $\ell$ in $\mathbb{F}^2$ be $\alpha$-rich in a Cartesian product $Y\times Y$ if … A set of lines is in general pos…

Let $N$ points be given in the Euclidean plane, and let a unit distance mean a pair of points whose Euclidean distance is $1$. The unit distance conjecture. The number of unit dist…

Let $(V,d)$ be a finite metric space with $n$ points. For distinct points $x,y,z\in V$, say that $z$ lies between $x$ and $y$ when … For distinct $x,y\in V$, define the line genera…

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a real line arrangement, and let $s$ be the number of its lines. Dirac–Motzkin conjecture. The inequality … ho…

Dirac's conjecture. There is a constant $c>0$ such that some point of $S$ is incident to at least $c n$ lines determined by $S$.

Let $d\geq 3$, let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements, and let a weak Nikodym set be a subset of $\mathbb{F}_q^d$ containi…

Oberlin's conjecture. Then

Let $I_C(m,n)$ denote the maximum number of incidences between $m$ points and $n$ circles. The circle-incidence conjecture. For some positive constant $c$, … This is a well-known i…

Let $0<u\leq1$ and $0\leq v\leq3$. A sine wave $(u,v)$-Furstenberg set or circular $(u,v)$-Furstenberg set is a set $F\subseteq\mathbb{R}^2$ of the type specified in the paper, wit…

Solymosi's conjecture. For any set of points $P_0$ and any set of lines $\mathcal{L}_0$ in the plane, the maximum number of incidences between $n$ points and $n$ lines in the plane…

Weak Dirac conjecture. Every set $\mathcal{P}$ of $n$ non-collinear points in the plane contains a point incident to at least

Supersolvable arrangement double-point conjecture. One has

Guth–Zahl conjecture. Let $p\geqslant\frac{n}{n-1}$. For every $\varepsilon>0$, there are a complexity $D=D_{\varepsilon,n}\in\mathbb{N}$ and a constant $C_{\varepsilon,n}>0$ such…

Let $\mathcal{P}\subset \mathbb{P}^{2}_{\mathbb{C}}$ be a finite set of mutually distinct $n$ points, and let $\mathcal{L}(\mathcal{P})$ be the set of lines determined by pairs of…

Let $\Gamma$ be a set of $L$ irreducible algebraic curves in $\mathbb{C}^n$, each of degree at most $d$, and let $P_2(\Gamma)$ denote the set of points incident to at least two cur…

Let $r\geq 2$, and let $\mathcal P\subset\mathbb{C}^d$ be a set of $n$ points. Write $\mathcal{L}_r(\mathcal P)$ for the set of lines incident to at least $r$ points of…

Let $k$ be a fixed positive integer, and let a family consist of $n$ pairwise intersecting bi-infinite $x$-monotone curves such that any two curves intersect at most $k$ times. Lin…

A cyclic $(n_k)$ configuration is a cyclic configuration in which every point lies on $k$ lines and every line contains $k$ points. Unsplittability conjecture. All cyclic $(n_k)$ c…

Multiplicity-line conjecture. If $B$ is small, then there is some $j\in\mathbb{F}_q\cup\{\infty\}$ such that every point $P\in B$ with $m_P\geq3$ lies on the line $\ell(j,b_j)$. Th…

Let $q$ be odd, and write $\ell(m,b)$ and $\ell(\infty,a)$ for the lines $y=mx+b$ and $x=a$ in $\mathbb{F}_q^2$. A Besicovitch set is a set $B\subset\mathbb{F}_q^2$ containing one…

Let $S$ be a finite set of points in the plane with rational coordinates. Write $\mathcal{T}(S)$ for the set of points that are intersections of two lines, each determined by a pai…

Let $0<\delta\ll1$, let $\Omega$ be a $\delta$-separated set of directions, and for each $\omega\in\Omega$ let $R_\omega$ be a $(\delta,1/2)_2$ set contained in a rectangle of dime…

Let $\Delta_{\mathrm H}(n)$ denote the largest possible minimum area of a triangle determined by $n$ points in the unit square. Improved Heilbronn bound conjecture. There exists an…