15 problems

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 $P$ be a finite set of $n$ points in the plane, and consider the most popular distance determined by pairs of points in $P$. Erdős's conjecture. For every $\epsilon>0$, the num…

Let $G=([n],E)$ be a graph with $|E|\gtrsim n^{7/6}$. Let ${\bf p}\in({\mathbb R}^2)^n$ be a realization of $G$ such that, for every vertex $v\in[n]$, the neighbours of $v$ are not…

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

Erdős unit distance conjecture. The number of unit distances determined by $P$ is at most

Let $n$ points be chosen in the Euclidean plane, and consider the maximum number of pairs at distance $1$ among them. An appropriately scaled $\sqrt{n}\times\sqrt{n}$ section of th…

Integer-lattice conjecture. For $d=2,3$, an appropriately scaled finite section of the integer lattice asymptotically maximizes the number of unit distances.

Let $u(n)$ denote the maximum number of pairs of points at unit distance among $n$ points in the Euclidean plane. Erdős's unit distances conjecture. The maximum number of unit dist…

Let $P$subset the plane be a set of $n$ points, and fix a Euclidean distance. The number of pairs of points in $P$ separated by that distance is at most a constant depending on…

Let $N$ be a large natural number, and let $x_1,\ldots,x_N$ be points in the plane. Erdős unit distances conjecture. The largest number of pairs … at unit distance is…

Non-tightness conjecture. The bound

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 a convex $n$-gon be given, and consider the distances between pairs of its vertices. The maximum number of pairs at unit distance is the maximum number of unit distances determ…

Let $u(n)$ denote the maximum number of pairs at unit distance determined by an $n$-point set in the Euclidean plane. Erdős proved the lower bound … for a constant $c>0$, attained…