349 problems

Erdős's distinct distances conjecture. The size of $\Delta E$ is nearly linear in terms of the size of $E$. Guth and Katz settled this problem in the plane, while the formulation c…

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…

Oberlin's conjecture. Then

Erdős–Falconer distance conjecture. If $|E|\gg q^m$, then $|\Delta_Q(E)|\gg q$.

Fourier decay conjecture.

Dirac's conjecture. For $|S|$ sufficiently large,

Let $\ell_1$ and $\ell_2$ be lines, and let $\mathcal P_1\subseteq\ell_1$ and $\mathcal P_2\subseteq\ell_2$ be sets of $n$ points. For point sets…

Wolff's finite-field Kakeya conjecture. There is a constant $c_n>0$, depending only on $n$, such that every Kakeya set $K\subset\mathbb{F}_q^n$ satisfies

Let $\mathcal{X}$ be a set of $n$ non-collinear points in the plane, and let $t(\mathcal{X})$ denote the maximum number of lines spanned by $\mathcal{X}$ that are incident with a p…

Finite-field Kakeya conjecture. Such a Kakeya set should have cardinality comparable to that of the ambient space, namely $|K|\approx|\mathbb{F}_q^n|=q^n$ up to a multiplicative co…

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 $\mathcal{E}$ be a finite subset of $\mathbb{R}^d$ with $d\geq 2$, and let $\Delta(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. Erdős's…

Polynomial Wolff axioms conjecture. The following assertions should hold:

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 $u(n)$ be the maximum number of unit distances determined by $n$ points in the plane. Erdős's unit distance conjecture. … Here the bound is attained, up to the stated order, by…

Let $P$ be a set of $n$ points in the Euclidean plane, and let … Write $X\lessapprox Y$ with controlling parameter $n$ when, for every $\epsilon>0$, there exists $C_\epsilon>0$ suc…

Let an incidence graph have $n$ points and $m$ lines, with edges representing incidences between points and lines. The de Caen–Székely conjecture. The maximum number of $6$-cycles…

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

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 $\mathbb{F}$ 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…

For an $n$-point set $P\subset\mathbb R^2$, let the distinct-distance number be the number of distinct Euclidean distances determined by pairs of points of $P$. Erdős's conjecture.…

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…

Finite-field Erdős distance conjecture. If $E$ has cardinality at least $Cq^{d/2}$, with $C$ sufficiently large, then

Let $f\in\mathbb{R}[x,y]$ be a fixed bivariate polynomial, and let $A,B\subset\mathbb{R}$ be finite sets with $|A|=|B|=n$. Call $f$ non-special if it is neither additive, meaning…

Ordered-group projective-plane conjecture. An ordered abelian group cannot locally trace define an infinite projective plane.