64 problems

Let $d\ge 2$ and let $A\subseteq \mathbb F_q^d$, where $q$ is an odd prime power. Define … The Erdős–Falconer distance exponent is the smallest $s>0$ such that $|A|\ge Cq^s$ implie…

Erdős's conjecture. There exists $N$ such that no crescent configurations of size $n$ exist for all $n\geq N$.

Let an integral circular point set be a finite set of points on a circle in the Euclidean plane such that the distance between every pair of points is an integer. Anning's conjectu…

Koh–Shen conjecture. If $|E||F|\gg_C q^d$, then

Conway and Sloane's conjecture. For all $N$ sufficiently large, an optimal set of $N$ points in $n$-dimensional space should be a subset of an $n$-dimensional lattice having minima…

The 3-connected Laman graph non-solvability conjecture. If a generic Laman graph is $3$-connected, then its constraint equations are not solvable by radicals (not-RS).

Spherical Erdős distance conjecture. On $S^2$, the Erdős distance conjecture should hold without the logarithmic factor; that is, $A$ should determine at least a constant times $N$…

Let $\mathbb F$ be a field of characteristic $0$. For each $n\geq 2$, let $D_n(\mathbb F)$ denote the set of positive distances that are forced by finite unit-distance configuratio…

Let $n\geq 2$, let $\mathbb F$ be a field of characteristic $0$, and let a map $f:\mathbb R^n\to\mathbb F^n$ preserve unit distance, meaning that $\varphi_n(f(x),f(y))=1$ whenever…

Let $E\subset\mathbb{R}^d$ be a compact set, and let … The Hausdorff dimension of $E$ equal to $1$ should suffice to ensure that $A_{\Delta}(E)$ has positive Lebesgue measure. Graf…

Large minimal-distance conjecture. There is a fixed constant $c>0$ such that the minimal distance is at least

Large minimal-distance conjecture. There is a fixed constant $c>0$ such that the minimal distance is at least

Let $\mathcal C$ be an $n$-point set in Euclidean space whose pairwise distances are mutually separated by at least $1$, and let $\operatorname{diam}(\mathcal C)$ denote its diamet…

Maximizer structure conjecture. The following properties hold:

Let $\mathcal{S}_{ij}$ be a Schubert variety in ${\rm Gr}(2,n)$. GD-degree conjecture. If $1<j-i<n-1$, then the Schubert variety $\mathcal{S}_{ij}$ has GD degree $2$. The paper rep…

Let $\mathcal{C}\subset\mathbb{P}^3$ be a curve, and let the Chow threefold and secant surface associated with $\mathcal{C}$ have Grassmannian-distance degree ${\rm GDdegree}$ and…

Let $\mathcal{C}\subset\mathbb{P}^3$ be a curve, and consider the associated Chow threefold, secant surface, and tangent curve of lines, together with a data line and an optimal li…

Let a surface in ${\rm Gr}(2,4)$ be defined by generic polynomials of degrees $d_1,d_2$, and let a threefold in ${\rm Gr}(2,4)$ be defined by a generic polynomial $f$ of degree $d$…

Perturbation conjecture. For every $\epsilon>0$, there is a set $S'=\{p'_1,p'_2,\ldots,p'_n\}$ in bijective correspondence with $S$ such that each corresponding pair satisfies…

For a configuration of $N$ points in ${\mathbb{R}}^3$, let $u(N)$ denote the largest number of pairs realizing the same distance. Let $C>0$ be a constant independent of $N$. Erdős…

For an $N$-point configuration in ${\mathbb{R}}^3$, let $d(N)$ denote the smallest number of distinct pairwise distances that can occur among its points. Here $c>0$ is a constant i…

Minimal-distance lower-bound conjecture. For all $N\geq 2$,

Let $f(t)=(t,t^2,\ldots,t^d)\in\mathbb{A}^d$, with $k\leq d$. Let $\varphi_{2,k}$ denote the map used to define the second-order osculating construction, let $Q\subseteq H_\infty$…

Let $d\ge 2$ be an even integer, let $A,B\subseteq\mathbb F_q^d$, and define … where $\|v\|=\sum_{i=1}^dv_i^2$. Two-set Erdős–Falconer conjecture. If … for a sufficiently large con…

Even-dimensional Erdős–Falconer conjecture. If $|A|\ge Cq^{d/2}$ for a sufficiently large constant $C$ independent of $q$, then