Related papers: Dense arrangements are locally very dense I

In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $…

According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at most $2^{n+O\left({n^{2/3}\log n}\right)}$…

Given a set of $s$ points and a set of $n^2$ lines in three-dimensional Euclidean space such that each line is incident the $n$ points but no $n$ lines are coplanar, then we have $s=\Omega(n^{11/4})$. This is the first nontrivial answer to…

In this short note we use the polynomial partitioning lemma to strengthen a recent result of Dvir and Gopi about the number of rich lines in high dimensional Euclidean spaces. Our result shows that if there are sufficiently many rich lines…

Let $p_0,\ldots,p_n$ be a finite sequence of points in an Euclidean space $\R^d$. Suppose that there is a (pointlike) particle sitting at each point $p_i$. In a ``legal'' move, any one of them can jump over another, landing on the other…

Exceptional points~(EPs) appear as degeneracies in the spectrum of non-Hermitian matrices at which the eigenvectors coalesce. In general, an EP of order $n$ may find room to emerge if $2(n-1)$ real constraints are imposed. Our results show…

The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_{\epsilon} n^{1+\epsilon}$ for any $\epsilon>0$. The best known…

A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its…

We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the…

We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a…

Given a finite set of points $C \subseteq \mathbb{R}^d$, we say that an ordering of $C$ is protrusive if every point lies outside the convex hull of the points preceding it. We give an example of a set $C$ of $5$ points in the Euclidean…

The Erd\H{o}s-Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that any…

Using recent developments on the theory of locally decodable codes, we prove that the critical size for Szemer\'edi's theorem with random differences is bounded from above by $N^{1-\frac{2}{k} + o(1)}$ for length-$k$ progressions. This…

The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek…

We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are $n$ balls of equal masses and radii 1, and at the time of a collision between…

We study a generalization of Erd\H os's unit distances problem to chains of $k$ distances. Given $\mathcal P,$ a set of $n$ points, and a sequence of distances $(\delta_1,\ldots,\delta_k)$, we study the maximum possible number of tuples of…

Say that a subset S of the plane is a "circle-center set" if S is not a subset of a line, and whenever we choose three noncollinear points from S, the center of the unique circle through those three points is also an element of S. A problem…

Let $p_1,p_2,p_3$ be three distinct points in the plane, and, for $i=1,2,3$, let $\mathcal C_i$ be a family of $n$ unit circles that pass through $p_i$. We address a conjecture made by Sz\'ekely, and show that the number of points incident…

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:…