Related papers: A de Bruijn - Erd\H{o}s theorem and metric spaces
As a variant of the celebrated Szemer\'edi--Trotter theorem, Guth and Katz proved that $m$ points and $n$ lines in $\mathbb{R}^3$ with at most $\sqrt{n}$ lines in a common plane must determine at most $O(m^{1/2}n^{3/4})$ incidences for…
Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is…
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…
Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P. This confirms, for large n, a conjecture of…
In this paper we obtain a new lower bound on the Erd\H{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb{F}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances…
We prove a version of the Erd\H{o}s--Beck Theorem from discrete geometry for fractal sets in all dimensions. More precisely, let $X\subset \mathbb{R}^n$ Borel and $k \in [0, n-1]$ be an integer. Let $\dim (X \setminus H) = \dim X$ for every…
Chen and Chv\'atal introduced the notion of lines in hypergraphs; they proved that every 3-uniform hypergraph with $n$ vertices either has a line that consists of all $n$ vertices or else has at least $\log_2 n$ distinct lines. We improve…
In 1951, Gabriel Dirac conjectured that every set P of n non-collinear points in the plane contains a point in at least n/2-c lines determined by P, for some constant c. The following weakening was proved by Beck and Szemer\'edi-Trotter:…
The paper is devoted to a proof of the de Bruijn-Erd\"os theorem in incidence geometry based on the Ph. Hall's marriage theorem (the theorem about the systems of distinct representatives).
We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this…
Erd\"os proved in 1946 that if a set $E\subset\mathbb{R}^n$ is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in $\mathbb{R}^n$ with the property that the nearest point in $E$ is not unique, can be…
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…
We prove that no ball admits a non-harmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erd\H os, which says that the number of distances determined by $n$ points in ${\Bbb R}^d$ is at least $C_d…
The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. 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 points in the…
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 extend the range of linear systems of plane curves for which it is known that the dimension is equal to the expected dimension. We use a degeneration technique due to C. Ciliberto and R. Miranda. From our result we deduce, as a…
In this article we address the problem of computing the dimension of the space of plane curves of degree $d$ with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension…
In 2008 Chen and Chv\'atal conjectured that any metric space on n points has at least n lines, unless all the points belong to one line. Chv\atal proved in 2014 that this is indeed the case for metric spaces with distances 0, 1 and 2. In…
The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…
The famous Szemer\'{e}di-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for…