154 problems

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 $k$ 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…

Let $K \subset \mathbb{R}^{2}$ be compact, let $\alpha \in[0,1]$, and let $\operatorname{vis}_{\sigma}(K)$ be the visible part of $K$ in direction $\sigma$. Exceptional-set estimat…

Let $K \subset \mathbb{R}^{2}$ be compact. For $\sigma \in[-1,1]$, let $\operatorname{vis}_{\sigma}(K)$ denote the points of $K$ that are not followed by another point of $K$ along…

Let $A\subset\mathbb{R}^2$ be a finite planar point set. Two points of $A$ are visible if the open segment between them contains no other point of $A$; a mutually visible subset is…

Let $g>0$. A simple tiling $\mathcal T$ of a genus-$g$ surface has an associated incidence theorem over a division ring ${\mathrm D}$, and $n$ denotes the dimension of the ambient…

Let $\mathcal T$ be a simple tiling of a surface of positive genus, meaning that its underlying graph is $(n+1)$-degenerate, and let ${\mathrm D}$ be a division ring. The incidence…

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…

Let $\mathcal{K}=\mathcal{K}_1\uplus\mathcal{K}_2$ be a finite 2-colored family of convex sets in $\mathbb{R}^3$. Suppose that every $K_1\in\mathcal{K}_1$ and $K_2\in\mathcal{K}_2$…

Oberlin's conjecture. Then

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…

Polynomial Wolff axioms conjecture. The following assertions should hold:

Strict realizability conjecture. If $G$ is a $K_{2,3}^{'}$-free graph which is independent in $\mathcal{R}_\ell$, then

Linear distinct-dot-products conjecture. For every such sequence,

Let $S(n,d)$ denote the number of $(d+2)$-tuples of points in $[n]^d$ that lie in a $(d-1)$-dimensional sphere. Sphere-counting conjecture. For any integer $d\geqslant 2$, … This c…

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 $n$ be a positive integer, and consider the $n\times n$ grid. Suppose it is covered by $m$ circles, with no two circles having equal radius. Distinct-radii circle-covering conj…

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 $\delta\in(0,1)$. Let $(L,Y)_\delta$ be a set of directional $\delta$-separated lines in $\mathbb{R}^n$ with an $(\varepsilon_1,\varepsilon_2,C_Y)$-two-ends, $\lambda$-dense sh…