55 problems

Are there, for all large $n$, some points $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is…

Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate $g_d(n)$. In particular, does…

Does there exist a sequence of finite sets $A_n\subseteq\mathbb R^2$ such that $|A_n|=n$, every subset of four points of $A_n$ determines at least three distinct distances, and the…

If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by $\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.$ Let $h(n)$ be the maximal number of un…

For a finite set $A\subset\mathbb R^2$, let $n=|A|$, let $D(A)$ be the set of distances determined by unordered pairs of distinct points of $A$, and let the multiplicity of a dista…

If $A$ is a finite set of $n$ distinct points in $\mathbb{R}^2$ in convex position, does $A$ determine at least $\left\lfloor n/2\right\rfloor$ distinct Euclidean distances?

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}$, where the points are ordered such that $R(x_1)\leq \cdots \leq R(x_n).$ Let…

Let $A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\}$ be a measurable set with no integer distances, that is, such that $\lvert a-b\rvert ot\in \mathbb{Z}$ for any distinct…

Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$…

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to other $x_i$?

Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\leq t$. For example $f(1)=6$,…

Let $S\subset\mathbb{R}^2$ be a set such that for all $x,y\in S$, $\operatorname{dist}(x,y)\ne1$. Must there exist four points $p_0,p_1,p_2,p_3\in\mathbb{R}^2\setminus S$ forming a…

Let $P$ be a finite set of points in $\mathbb{R}^2$ whose points are in convex position. Let $\{u_1,\ldots,u_t\}$ be the set of distances determined by pairs of points of $P$, and…

For fixed $d≥2$, let $F_d(n)$ be the largest integer such that every set of $n$ points in $R^d$ contains $F_d(n)$ points whose pairwise distances are all distinct. Determine the or…

Let $P_d(n)$ be the largest integer such that every set of $n$ points in $R^d$ contains $P_d(n)$ points with no three forming an isosceles triangle. Estimate $P_d(n)$; in particula…

For each $d≥1$, can $R^d$ be partitioned into countably many sets such that within each set every positive distance occurs for at most one unordered pair of points?

Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are th…

Let $g(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$.

Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $f_d(n)$.

Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\mathbb{R}^d$, all of distance at least $1$ apart, there are at most $f_d(n)$ many pairs of points which are d…

Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular…

(i) Let $A\subset\mathbb R^2$ be a finite set with no three points on a line. Must $A$ determine at least $\lfloor |A|/2\rfloor$ distinct distances? (ii) No. The following assertio…

Let $n\ge3$ distinct points $p_0,\ldots,p_{n-1}\in\mathbb R^2$ form a convex polygon. Must there exist a vertex $p_i$ having at least $\lfloor n/2\rfloor$ distinct distances from t…

Let $A$ be a set of $n$ distinct points in $\mathbb{R}^2$. If $s_{\min}$ and $s_{\max}$ are the numbers of unordered pairs of points attaining respectively the smallest and largest…

Let $A\subset \mathbb{R}$ be a set of size $n$ such that every subset $B\subseteq A$ with $\lvert B\rvert =4$ has $\lvert B-B\rvert\geq 11$. Find the best constant $c>0$ such that…