55 problems

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…

Let $n$ points form a convex polygon, and for each point count the distinct distances it determines to the other points. Erdős's convex-polygon pinned-distance conjecture. There is…

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…

Erdős–Purdy–Straus conjecture. This construction is tight for all sufficiently large $n$. The conjecture asks whether the displayed upper bound gives the exact minimum once $n$ is…

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 $h_1(n)$ be the maximum integer $t$ such that every set of $n$ points in $\mathbb{R}$ contains a subset $S$ of $t$ points for which all $\binom{t}{2}$ pairwise distances are di…

In the Euclidean plane, let $N$ points be given, and consider the least number of distinct distances determined by any such configuration. Erdős's distinct distances conjecture. Th…

Let $A$ be an $N$-point set in the real plane. Erdős's distinct distances conjecture. The set $A$ should determine at least … distinct distances, for some universal constant $c>0$.…

Erdős–Lovász–Vesztergombi conjecture. The number of distinct such distances is linear in $n$.

A $d$-norm is a norm $\|\cdot\|$ on $\mathbb{R}^d$, identified with its unit ball. A set of points is full-dimensional when it is not contained in an affine hyperplane, and a set o…

For $d\geqslant 3$, consider finite sets of $n$ points in Euclidean $\mathbb{R}^d$, and count the distinct values of the Euclidean distance between pairs of points. The Euclidean d…

Minimum-degree-two realisation conjecture. One has

Graphical distinct-realisation conjecture. For every $\varepsilon>0$, there exists a positive constant $C_\varepsilon$, independent of $P$, such that

Weak pinned distance conjecture. For every $\varepsilon>0$, there exists a constant $C_\varepsilon>0$ such that every finite point set $P\subseteq\mathbb R^2$ contains a point…

Erdős's distinct distances conjecture. The correct answer is

Let $\ell_1$ and $\ell_2$ be two lines, and let $\mathcal P_1\subseteq\ell_1$ and $\mathcal P_2\subseteq\ell_2$ be sets of $n$ points. Define … to be the set of distances between t…

Erdős's structural conjecture. If $P$ defines

Erdős's distinct distances conjecture. For any $P\subset\mathbb R^d$,

Let $P$ be a finite subset of $\mathbb{R}^d$, where $d\geq 2$, with $|P|=N$. Define … where $|x|=\sqrt{x_1^2+x_2^2+\dots+x_d^2}$ is the usual Euclidean norm. Erdős's distinct-dista…

Distinct-distances conjecture. For every such pair of sets, $P_1\times P_2$ spans $\Omega(n^{3/2})$ distinct distances, unless each of $\gamma_1$ and $\gamma_2$ is an algebraic hel…

Let $D_{\|.\|}(n)$ denote the minimum possible number of distinct distances determined by $n$ points in $\mathbb{R}^d$ according to a $d$-norm $\|.\|$. Brass's conjecture. For ever…

Let $P$ be a finite set in $\mathbb{R}^m$, and let $E_2(P)$ denote its second distance energy, namely the number of ordered quadruples $(a,b,c,d)\in P^4$ satisfying $|a-b|=|c-d|$.…

Consider a configuration of $n$ points on a sphere, and let its distinct distances mean the distinct distances among pairs of points in the configuration. Spherical distinct-distan…

Let $g(n)$ be the minimum number of distinct distances determined by a set of $n$ points in the plane. Erdős's lattice conjecture. The $\sqrt n\times\sqrt n$ lattice, which yields…

Let $k\geq 2$ points be arranged in the plane, and consider the number of distinct pairwise distances among them. Erdős's distinct distances conjecture. The number of distinct dist…