Related papers: On the Number of Discrete Chains

Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of…

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of…

Let P_1 and P_2 be two sets of points in the plane, so that P_1 is contained in a line L_1, P_2 is contained in a line L_2, and L_1 and L_2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2…

We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the…

We survey the variants of Erd\H{o}s' distinct distances problem and the current best bounds for each of those.

We begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the following question: given $k\in \mathbb{N}$, what is the maximum number of points in a plane that determine at most $k$ distinct distances, and can such optimal…

We study the dual variants of the Erd\H{o}s's distinct distances and unit distance problems. Instead of considering distances determined by points, we consider simplex volumes determined by hyperplanes. We investigate: (1) the maximum…

In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset…

We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,...,n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to…

We consider the number of distinct distances between two finite sets of points in ${\bf R}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary…

Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance $n$-chains defined as $$ \Delta_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show that for $2\leq n=O_{|P|}(1)$ we have $$|\Delta_n(P)|\gtrsim…

The Erd\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $\Theta(N/\sqrt{\log(N)})$…

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set $S$ of $s$ points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in…

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1…

A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on…

The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear…

We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of…

Let $p_1,p_2,p_3$ be three non-collinear points in the plane, and let $P$ be a set of $n$ other points in the plane. We show that the number of distinct distances between $p_1,p_2,p_3$ and the points of $P$ is $\Omega(n^{6/11})$, improving…