Related papers: Incidences between points and lines on a two-dimen…

The Szemer\'edi-Trotter theorem gives a bound on the maximum number of incidences between points and lines on the Euclidean plane. In particular it says that $n$ lines and $n$ points determine $O(n^{4/3})$ incidences. Let us suppose that an…

We bound the number of incidences between points and spheres in finite vector spaces by bounding the sum of the number of points in the pairwise intersections of the spheres. We obtain new incidence bounds that are interesting when the…

It is shown that $n$ points and $e$ lines in the complex Euclidean plane ${\mathbb C}^2$ determine $O(n^{2/3}e^{2/3}+n+e)$ point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemer\'edi and…

Two spheres with centers $p$ and $q$ and signed radii $r$ and $s$ are said to be in contact if $|p-q|^2 = (r-s)^2$. Using Lie's line-sphere correspondence, we show that if $F$ is a field in which $-1$ is not a square, then there is an…

Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime. Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$ determine…

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is…

We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not…

A classical result by Erd\H{o}s, and later on by Bondy and Simonivits, states that every $n$-vertex graph with no cycle of length $2k$ has at most $O(n^{1+1 /k})$ edges. This bound is known to be tight when $k \in \{2,3,5\},$ but it is a…

This article is concerned with the rigidity properties of geometric realizations of incidence geometries of rank two as points and lines in the Euclidean plane; we care about the distance being preserved among collinear points. We discuss…

Let $d$ and $k$ be integers with $1 \leq k \leq d-1$. Let $\Lambda$ be a $d$-dimensional lattice and let $K$ be a $d$-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of $k$-dimensional…

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also…

We prove new incidence bounds between a plane point set, which is a Cartesian product, and a set of translates $H$ of the hyperbola $xy=\lambda\neq 0$, over a field of asymptotically large positive characteristic $p$. They improve recent…

In this paper we prove, for all $d \ge 2$, that for no $s<\frac{d+1}{2}$ does $I_s(\mu)<\infty$ imply the canonical Falconer distance problem incidence bound, or the analogous estimate where the Euclidean norm is replaced by the norm…

The aim of this paper is to obtain a classification of the scrolls in Pn which are defined by a one-dimensional family of lines meeting a certain set of linear spaces in Pn, a first classification for genus 0 and 1 is given in paper [1].…

In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it…

We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…

Let $\mathcal{P}$ be a set of $m$ points and $\mathcal{L}$ a set of $n$ lines in $K^2$, where $K$ is a field with char$(K)=0$. We prove the incidence bound $$\mathcal{I}(\mathcal{P},\mathcal{L})=O(m^{2/3}n^{2/3}+m+n).$$ Moreover, this bound…

We consider configurations of lines in 3-space with incidences prescribed by a graph. This defines a subvariety in a product of Grassmannians. Leveraging a connection with rigidity theory in the plane, for any graph, we determine the…

Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are…

We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we…