Related papers: On cliques in three-dimensional dense point-line a…
Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually…
Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a constraint…
Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoi\v{c}i\'{c} (2003) showed that the limiting set is dense…
This is an incomplete attempt to show that the upper bound of $\lesssim n^\frac{4}{3}$ on the number unit distances determined by a large finite set of $n$ points in the plane is not sharp. The methods also say something about sets of $n$…
We generalize the Szemer\'edi-Trotter incidence theorem, to bound the number of complete \emph{flags} in higher dimensions. Specifically, for each $i=0,1,\ldots,d-1$, we are given a finite set $S_i$ of $i$-flats in $\R^d$ or in $\C^d$, and…
A joint of a set of lines $\mathcal{L}$ in $\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz…
A classical theorem of De Bruijn and Erd\H{o}s asserts that any noncollinear set of n points in the plane determines at least n distinct lines. We prove that an analogue of this theorem holds for graphs. Restricting our attention to…
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…
It is shown that the number of distinct types of three-point hinges, defined by a real plane set of $n$ points is $\gg n^2\log^{-3} n$, where a hinge is identified by fixing two pair-wise distances in a point triple. This is achieved via…
We completely characterize point--line configurations with $\Theta(n^{4/3})$ incidences when the point set is a section of the integer lattice. This can be seen as the main special case of the structural Szemer\'edi-Trotter problem. We also…
We prove a special case of Erd\H{o}s' unit distance problem using a corollary of the subspace theorem bounding the number of solutions of linear equations from a multiplicative group. We restrict our attention to unit distances coming from…
We show that the number of unit distances determined by n points in R^3 is O(n^{3/2}), slightly improving the bound of Clarkson et al. established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth…
We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…
The following generalisation of the Erd\H{o}s unit distance problem was recently suggested by Palsson, Senger and Sheffer. Given $k$ positive real numbers $\delta_1,\dots,\delta_k$, a $(k+1)$-tuple $(p_1,\dots,p_{k+1})$ in $\mathbb{R}^d$ is…
A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been…
For an integer $r \ge 3$ and a subset $L \subset [0,r-1]$, a graph $G$ is $(K_{r}, L)$-intersecting if the number of vertices in the intersection of every pair of $K_r$ in $G$ belongs to $L$. We study the maximum number of $K_r$ in an…
The Erd\H{o}s--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs…
We consider the problem of finding a large clique in an Erd\H{o}s--R\'enyi random graph where we are allowed unbounded computational time but can only query a limited number of edges. Recall that the largest clique in $G \sim G(n,1/2)$ has…
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…
We give a new bound on the parameter $\lambda$ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph $G$, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014).…