Csaba D. Toth
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 Szemerédi and Trotter about point-line incidences in the real Euclidean plane ${\mathbb R}^2$.
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We establish a Szemeredi-Trotter type result for hyperbolas in $\mathbb F_p\times\mathbb F_p$
We study a Szemerédi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.
Structure of cell decompositions in Extremal Szemerédi-Trotter examples
本文提出一个由O(N^{1/3})个参数描述的构造方法，用于生成接近极值的点线配置，并证明任何近极值例子都与该方法的成功实例密切相关。
Let $q$ be a power of a prime and $\mathbb{F}_q$ the finite field consisting of $q$ elements. We prove explicit upper bounds on the number of incidences between lines and Cartesian products in $\mathbb{F}_q^2$. We also use our results on point-line incidences to give new sum-product type estimates concerning sums of reciprocals.
Multi-parameter Szemerédi-Trotter-type theorems and applications in finite fields
本文在有限域向量空间中建立了优于一般高维结果的多参数点-线关联估计，并展示了其在几何与数论组合问题中的多种应用。
A group-action Szemerédi-Trotter theorem and applications to orchard problems in all characteristics
本文在任意域上证明群作用版 Szemerédi-Trotter 定理，并将其用于有限域和复数域中可约三次曲面的共线三元组计数，改进已有界。
Structural Szemerédi-Trotter for Lattices and their Generalizations
本文研究整数格截面及其推广中的点线关联结构，完整刻画 Θ(n^{4/3}) 关联构型，并利用乘法能量对等差数列笛卡尔积等情形给出部分结论。
We present a new family of sharp examples for the Szemerédi-Trotter theorem. These are the first examples not based on a rectangular lattice. We also include an application to the discrete inverse Loomis-Whitney problem.
Sharp Szemerédi-Trotter constructions from arbitrary number fields
该研究借鉴Elekes的思路，用任意数域的基构造任意高维GAP的笛卡尔积，得到无穷多精确Szemerédi-Trotter构造，拓展了Guth与Silier的相关结果且简化了分析。
The Szemerédi-Trotter theorem over arbitrary field of characteristic zero
本文利用Baby Lefschetz原理将问题归约到复数情形，证明了特征零任意域上的Szemerédi-Trotter关联界，并推导出Beck定理、和积估计等推广结果。
We study configurations of $n$ points and $n$ lines that form $Θ(n^{4/3})$ incidences, when the point set is a Cartesian product. We prove structural properties of such configurations, such that there exist many families of parallel lines or many families of concurrent lines. We show that the line slopes have multiplicative structure or that many sets of $y$-intercepts have additive structure. We introduce the first infinite family of configurations with $Θ(n^{4/3})$ incidences. We also derive a new variant of a different structural point-line result of Elekes. Our techniques are based on the …
We prove a Szemerédi-Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and Sárközy on the solvability of the equations $a + b = cd$ and $ab + 1 = cd$ over a finite field. Other analogous results that are known to hold in finite fields are generalized to finite quasifields.
We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemerédi-Trotter type theorem for 2--planes in $\mathbb{R}^4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemerédi-Trotter type theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem …
We prove an analog of the Szemerédi-Trotter theorem in the plane for definable curves and points in any o-minimal structure over an arbitrary real closed field $\mathrm{R}$. One new ingredient in the proof is an extension of the well known crossing number inequality for graphs to the case of embeddings in any o-minimal structure over an arbitrary real closed field.
We generalize the Szemerédi-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 (complete) flag is a tuple $(f_0,f_1,\ldots,f_{d-1})$, where $f_i\in S_i$ for each $i$ and $f_i\subset f_{i+1}$ for each $i=0,1,\ldots,d-2$. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in $\R^3$ such that a…
We estimate the number of incidences in a configuration of $m$ lines and $n$ points in dimension 3. The main term is $mn^{1/3}$ if we work over the real or complex numbers but $mn^{2/5}$ over finite fields. Both of these are optimal, aside from a multiplicative constant that is at most 5.
Two proofs and a tight Szemerédi--Trotter theorem
本文为Lewko与Miao–Xie分别提出的两个Szemerédi–Trotter定理推广证明提供几何直观并比较，利用Hasse导数改进证明，最终得到任意代数闭域上的紧关联对数量上界。
The Szemerédi-Trotter Estimate in Finite Field with its Applications
本文通过新多项式分解定理证明有限域上的锐利Szemerédi-Trotter估计，并应用于Furstenberg集估计、和积估计及傅里叶限制估计的改进。
We improve the well-known Szemerédi-Trotter incidence bound for proper 3--dimensional point sets (defined appropriately)
A Szemerédi-Trotter Theorem in Arbitrary Fields
本文在任意特征域上证明Szemerédi-Trotter定理，给出点线关联数的上界，并应用于抛物面延拓估计、Bourgain提取器及sum-product估计的改进。
On cliques in three-dimensional dense point-line arrangements
本文研究了三维空间中趋向 Guth--Katz 关联上界的极值稠密点线排列，证明了在 $m \ll n$ 时该排列必然包含处于一般位置的 $k$-团，这是对平面上 Szemerédi--Trotter 极值构造结果的类比。
We prove analogues of the Szemerédi-Trotter theorem and other incidence theorems using $δ$-tubes in place of straight lines, assuming that the $δ$-tubes are well-spaced in a strong sense.
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 Radoičić (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the Szemerédi-Trotter Theorem and an analysis of the ``minimum degree'' of the growing configuration.
We study a lower bound for the constant of the Szemerédi-Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies $I({\mathcal P},{\mathcal L})\ge (c+o(1)) |{\mathcal P}|^{2/3}|{\mathcal L}|^{2/3}$, with $c\approx 1.27$. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository perspective, this is the first full analysis of the constant of Erd\H os's construction.
We study the $δ$-discretized Szemerédi-Trotter theorem and Furstenberg set problem. We prove sharp estimates for both two problems assuming tubes satisfy some spacing condition. For both two problems, we construct sharp examples that have many common features.
Exploration on Incidence Geometry and Sum-Product Phenomena
本论文研究Erdős-Szemerédi猜想与关联几何的关系，将实数域上的和积估计技术推广到有限域，并在环Z/p^3Z上获得平分线数量的估计结果。
Recently Guth and Katz \cite{GK2} invented, as a step in their nearly complete solution of Erdős's distinct distances problem, a new method for partitioning finite point sets in $\R^d$, based on the Stone--Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi--Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generaliza…
The Szemerédi-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 arrangement of $n$ lines and $n$ points defines $cn^{4/3}$ incidences, for a given positive $c.$ It is widely believed that such arrangements have special structure, but no results are known in this direction. Here we show that for any natural number, $k,$ one can find $k$ points of the arrangement in general position such that any pair of them is incident to…
Given a set of $n$ points in $R^2$, the Szemerédi-Trotter theorem establishes that the number of lines which can be incident to at least $k > 1$ of these points is $O(n^2/k^3 + n/k)$. J.\ Solymosi conjectured that if one requires the points to be in a grid formation and the lines to be in general position---no two parallel, no three meeting at a point---then one can get a much tighter bound. We prove: for every $ε> 0$ there exists some $δ> 0$ such that for sufficiently large values of $n$, every set of lines in general position, each intersecting an $n \times n$ grid of points in at least $n^{…
On the structure of extremal point-line arrangements
本文证明达到 Szemerédi-Trotter 极值关联数的点线排列具有刚性：仅固定依赖常数 C 的少量点，即可确定排列的正比例部分。
The famous Szemerédi-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 Turán-type result for point-line incidence. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be two sets of $t$ lines in the plane and let $P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\}$ be the set of intersection points between $\mathcal{L}_1$ and $\mathcal{L}_2$. We say that $(P, \mathcal{L}_1 \cup \mathcal{L}_2)$ forms a \emph{natural $t\times t$ grid} if $|P| =t^2$,…
We establish improved finite field Szemeredi-Trotter and Beck type theorems. First we show that if P and L are a set of points and lines respectively in the plane F_p^2, with |P|,|L| \leq N and N<p, then there are at most C_1 N^{3/2-1/662+o(1)} incidences between points in P and lines in L. Here C_1 is some absolute constant greater than 1. This improves on the previously best-known bound of C_1 N^{3/2-1/806+o(1)}. Second we show that if P is a set of points in \mathbb{F}_p^2 with |P|<p then either at least C_2|P|^{1-o(1)} points in P are contained in a single line, or P determines least C_2 |…
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 Szemerédi-Trotter: every set P of n non-collinear points contains a point in at least n/c lines determined by P, for some large unspecified constant c. We prove that every set P of n non-collinear points contains a point in at least n/37 lines determined by P. We also give the best known constant for Beck's Theorem, proving that every set of n points with at most k collinear deter…
On forbidden configurations in point-line incidence graphs
本文否证了关于点线关联图中禁止构型的两个猜想，并引入新方法证明避免特定子构型时关联数的改进上界。
We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving $p$ if $p>…
The point-plane incidence theorem states that the number of incidences between $n$ points and $m\geq n$ planes in the projective three-space over a field $F$, is $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear points, with the extra condition $n< p^2$ if $F$ has characteristic $p>0$. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in $F^2$, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in $…
A bipartite graph $G$ is semi-algebraic in $\mathbb{R}^d$ if its vertices are represented by point sets $P,Q \subset \mathbb{R}^d$ and its edges are defined as pairs of points $(p,q) \in P\times Q$ that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in $2d$ coordinates. We show that for fixed $k$, the maximum number of edges in a $K_{k,k}$-free semi-algebraic bipartite graph $G = (P,Q,E)$ in $\mathbb{R}^2$ with $|P| = m$ and $|Q| = n$ is at most $O((mn)^{2/3} + m + n)$, and this bound is tight. In dimensions $d \geq 3$, we show that all such semi-algeb…
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 will discuss are: (1) Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc..), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the Szemeredi-Trotter theorem, over the reals and over finite fields and discuss…
We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\ldots,v_n \in \mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemerédi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. This …
Partitioning Theorems for Sets of Semi-Pfaffian Sets, with Applications
本文提出对半皮夫金集合族的多项式划分定理，并应用该定理推导了皮夫金版本的Szemerédi-Trotter定理及皮夫金曲线的关节数量界。
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