Search

Let $H_n^{(3)}$ be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, {\em wicket}, is formed by three rows and two columns of a $3 \times 3$ point matrix. In this note, we give a new lower bound on the Tur\'an number of wickets using estimates on cap sets. We also show that this problem is closely connected to important questions in additive combinatorics., Comment: An updated version, correcting a gap in the previous version

We describe constructions of infinite graphs which are not representable as integral graphs in the plane, addressing a question of Erd\H{o}s. We also mention some related problems.

In these notes, we consider a Tur\'an-type problem in hypergraphs. What is the maximum number of edges if we forbid a subgraph? Let $H_n^{(3)}$ be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, called {\em wicket}, is formed by three rows and two columns of a $3 \times 3$ point matrix. We describe two linear hypergraphs -- both containing a wicket -- that if we forbid either of them in $H_n^{(3)}$, then the hypergraph is sparse, and the number of its edges is $o(n^2)$. This proves a conjecture of Gy\'arf\'as and S\'ark\"ozy.

Let $\mathcal{P}$ be a set of points in the plane, and $\mathcal{S}$ a strictly convex set of points. In this note, we show that if $\mathcal{P}$ contains many translates of $\mathcal{S}$, then these translates must come from a generalized arithmetic progression of low dimension. We also discuss an application to the unit distance conjecture.

We prove a new Elekes-Szab\'o type estimate on the size of the intersection of a Cartesian product $A\times B\times C$ with an algebraic surface $\{f=0\}$ over the reals. In particular, if $A,B,C$ are sets of $N$ real numbers and $f$ is a trivariate polynomial, then either $f$ has a special form that encodes additive group structure (for example $f(x,y,x) = x + y - z$), or $A \times B\times C \cap\{f=0\}$ has cardinality $O(N^{12/7})$. This is an improvement over the previously bound $O(N^{11/6})$. We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent $3/2$. This has applications to questions in combinatorial geometry related to the Erd\H{o}s distinct distances problem. Like previous approaches to the problem, we rephrase the question as a $L^2$ estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples., Comment: 7 pages, 0 figures

We show that there are five types of planar curves such that arrangements of its translates are combinatorially equivalent to an arrangement of lines. These curves can be used to define norms giving constructions with many unit distances among points in the plane.

It is conjectured that if a finite set of points in the plane contains many collinear triples then there is some structure in the set. We are going to show that under some combinatorial conditions such pointsets contain special configurations of triples, proving a case of Elekes' conjecture. Using the techniques applied in the proof we show a density version of Jamison's theorem. If the number of distinct directions between many pairs of points of a pointset in convex position is small, then many points are on a conic.

Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a slightly better estimate in the real case. Another one is a geometric application. Further results of a similar type are given for normal and almost symmetric matrices.

We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb{R}^d$ generate high rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets.

It was proved by Ron Graham and the second author that for any coloring of the $N \times N$ grid using fewer than $\log \log N$ colours, one can always find a monochromatic isosceles right triangle, a triangle with vertex coordinates $(x, y),(x + d, y),$ and $(x, y + d).$ In this paper we are asking questions where not only axis-parallel, but tilted isosceles right triangles are considered as well. Both colouring and density variants of the problem will be discussed., Comment: Submitted to the Ron Graham Memorial Volume of INTEGERS

We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion., Comment: 16 pages, 3 figures

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the constant multiplier. We give a new proof to this result using bounds on crossing numbers of geometric graphs. We construct examples showing the limits of possible improvements. In particular, we show that there are arbitrarily large sets with different consecutive differences and sub-quadratic sumset sizes., Comment: To appear in Canad. Math. Bull

This note is a continuation of an earlier paper by the authors. We describe improved constructions addressing a question of Erd\H{o}s and Szemer\'edi on sums and products of real numbers along the edges of a graph. We also add a few observations about related versions of the problem.

We prove an extension of the Thue-Vinogradov Lemma and show some applications. This paper is another example for the application of the polynomial method., Comment: This the journal submission form

In this paper we show examples for applications of the Bombieri-Lang conjecture in additive combinatorics, giving bounds on the cardinality of sumsets of squares and higher powers of integers. Using similar methods we give bounds on the sum-product problem for matchings., Comment: 14 pages

We study arrangements of intervals in $\mathbb{R}^2$ for which many pairs form trapezoids. We show that any set of intervals forming many trapezoids must have underlying algebraic structure, which we characterise. This leads to some unexpected examples of sets of intervals forming many trapezoids, where an important role is played by degree 2 curves., Comment: 14 pages, 11 figures

We prove that the number of directions contained in a set of the form $A \times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| - \min\{|A|,|B|\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and $|A||B|

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to the constant., Comment: This paper supersedes arXiv:math/0503069

While the problem of determining whether an embedding of a graph $G$ in $\mathbb{R}^2$ is {\it infinitesimally rigid} is well understood, specifying whether a given embedding of $G$ is {\it rigid} or not is still a hard task that usually requires ad hoc arguments. In this paper, we show that {\it every} embedding (not necessarily generic) of a dense enough graph (concretely, a graph with at least $C_0n^{3/2}\log n$ edges, for some absolute constant $C_0>0$), which satisfies some very mild general position requirements (no three vertices of $G$ are embedded to a common line), must have a subframework of size at least three which is rigid. For the proof we use a connection, established in Raz [Ra], between the notion of graph rigidity and configurations of lines in $\mathbb{R}^3$. This connection allows us to use properties of line configurations established in Guth and Katz [GK2]. In fact, our proof requires an extended version of Guth and Katz result; the extension we need is proved by J\'anos Koll\'ar in an Appendix to our paper. We do not know whether our assumption on the number of edges being $\Omega(n^{3/2}\log n)$ is tight, and we provide a construction that shows that requiring $\Omega(n\log n)$ edges is necessary.

The Brown-Erd\H{o}s-S\'{o}s conjecture, one of the central conjectures in extremal combinatorics, states that for any integer $m\geq 6,$ if a 3-uniform hypergraph on $n$ vertices contains no $m$ vertices spanning at least $m-3$ edges, then the number of edges is $o(n^2).$ We prove the conjecture for triple systems coming from finite abelian groups.

We show the existence of infinitesimally rigid bipartite unit-bar frameworks in $\mathbb{R}^d$. We also construct unit-bar frameworks with girth up to 12 that are infinitesimally rigid in the plane. This answers problems proposed by Maehara., Comment: Python code included in ancillary files

In their seminal paper Erd\H{o}s and Szemer\'edi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erd\H{o}s-Szemer\'edi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets and ratio sets along the edges of graphs.

If $M$ is a set of nonsingular $k\times k$ matrices then for many pairs of matrices, $A,B\in M,$ the sum is nonsingular, $\det(A+B)\neq 0.$ We prove a more general statement on nonsingular sums with an application.

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and non-residues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant probability of such a high difference and show the existence of sets with an exceptionally large discrepancy.

The boundedness of the kissing numbers of convex bodies has been known to Hadwiger for long. We present an application of it to the sum-product estimate $$\max(|\mathcal{A}+\mathcal{A}|,|\mathcal{A}\mathcal{A}|)\gg\dfrac{|\mathcal{A}|^{4/3}}{\lceil\log|\mathcal{A}|\rceil^{1/3}}$$ for finite sets $\mathcal{A}$ of quaternions and of a certain family of well-conditioned matrices.

Two hexagons in the space are said to intersect heavily if their intersection consists of at least one common vertex as well as an interior point. We show that the number of hexagons on n points in 3-space without heavy intersections is o(n^2), under the assumption that the hexagons are "fat".

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)} $ it contains a collinear triple. Another application studies epsilon-nets in a point-line system in the plane. We prove the existence of some geometric constructions with a new tool, the so-called Hypergraph Container Method.

Let $P$ be a finite point set in the plane. A \emph{$c$-ordinary triangle} in $P$ is a subset of $P$ consisting of three non-collinear points such that each of the three lines determined by the three points contains at most $c$ points of $P$. Motivated by a question of Erd\H{o}s, and answering a question of de Zeeuw, we prove that there exists a constant $c>0$ such that $P$ contains a $c$-ordinary triangle, provided that $P$ is not contained in the union of two lines. Furthermore, the number of $c$-ordinary triangles in $P$ is $\Omega(|P|)$., Comment: 5 pages, now an estimate on c is given

Design matrices are sparse matrices in which the supports of different columns intersect in a few positions. Such matrices come up naturally when studying problems involving point sets with many collinear triples. In this work we consider design matrices with block (or matrix) entries. Our main result is a lower bound on the rank of such matrices, extending the bounds proved in {BDWY12,DSW12} for the scalar case. As a result we obtain several applications in combinatorial geometry. The first application involves extending the notion of structural rigidity (or graph rigidity) to the setting where we wish to bound the number of `degrees of freedom' in perturbing a set of points under collinearity constraints (keeping some family of triples collinear). Other applications are an asymptotically tight Sylvester-Gallai type result for arrangements of subspaces (improving {DH16}) and a new incidence bound for high dimensional line/curve arrangements. The main technical tool in the proof of the rank bound is an extension of the technique of matrix scaling to the setting of block matrices. We generalize the definition of doubly stochastic matrices to matrices with block entries and derive sufficient conditions for a doubly stochastic scaling to exist., Comment: Published in Discrete Analysis

Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper.

Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity., Comment: 10 pages, 4 figures

We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves are tangent. In particular, if no three curves are mutually tangent at a common point, then there are $O(n^{3/2})$ curve-curve tangencies. Second, given a family of algebraic plane curves and a set of $n$ curves from this family, we show that either there are $O(n^{3/2})$ points where two or more curves are orthogonal, or the family of curves has certain special properties. We obtain these bounds by transforming the arrangement of plane curves into an arrangement of space curves so that tangency (or orthogonality) of the original plane curves corresponds to intersection of space curves. We then bound the number of intersections of the corresponding space curves. For the case of curve-curve tangency, we use a polynomial method technique that is reminiscent of Guth and Katz's proof of the joints theorem. For the case of orthogonal curve intersections, we employ a bound of Guth and the third author to control the number of two-rich points in space curve arrangements., Comment: 22 pages, 1 figure. v4: final version; to appear in Discrete Analysis

The main result of this paper is that for any $c>0$ and for large enough $n$ if the number of edges in a 3-uniform hypergraph is at least $cn^2$ then there is a core (subgraph with minimum degree at least 2) on at most 15 vertices. We conjecture that our result is not sharp and 15 can be replaced by 9. Such an improvement seems to be out of reach, since it would imply the following case of a long-standing conjecture by Brown, Erd\H os, and S\'os; if there is no set of 9 vertices that span at least 6 edges of a 3-uniform hypergraph then it is sparse., Comment: 10 pages, 4 figures

We prove bounds on the number of incidences between a set of algebraic curves in $\mathbb{C}^2$ and a Cartesian product $A\times B$ with finite sets $A,B\subset \mathbb{C}$. Similar bounds are known under various conditions, but we show that the Cartesian product assumption leads to a simpler proof. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from $\mathbb{R}$ to $\mathbb{C}$. The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz., Comment: Many minor changes

We give bounds on the number of distinct differences $N_a-a$ as $a$ varies over all elements of a given finite set $A$, and $N_a$ is a nearest neighbour to $a$.

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 to a circle of each family is $O(n^{11/6})$, improving an earlier bound for this problem due to Elekes, Simonovits, and Szab\'o [Combin. Probab. Comput., 2009]. The problem is a special instance of a more general problem studied by Elekes and Szab\'o [Combinatorica, 2012] (and by Elekes and R\'onyai [J. Combin. Theory Ser. A, 2000])., Comment: 23 pages

In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\Omega(n^{4/3})$, for every pair of finite sets $A,B\subset{\mathbb R}$, with $|A|=|B|=n$ (where the constant of proportionality depends on ${\rm deg} f$), or else $f$ must be of one of the special forms $f(u,v)=h(\varphi(u)+\psi(v))$, or $f(u,v)=h(\varphi(u)\cdot\psi(v))$, for some univariate polynomials $\varphi,\psi,h$ over ${\mathbb R}$. This significantly improves a result of Elekes and R\'onyai (2000). Our results are cast in a more general form, in which we give an upper bound for the number of zeros of $z=f(x,y)$ on a triple Cartesian product $A\times B\times C$, when the sizes $|A|$, $|B|$, $|C|$ need not be the same; the upper bound is $O(n^{11/6})$ when $|A|=|B|=|C|=n$, where the constant of proportionality depends on ${\rm deg} f$, unless $f$ has one of the aforementioned special forms. This result provides a unified tool for improving bounds in various Erd\H os-type problems in geometry and additive combinatorics. Several applications of our results to problems of these kinds are presented. For example, we show that the number of distinct distances between $n$ points lying on a constant-degree parametric algebraic curve which does not contain a line, in any dimension, is $\Omega(n^{4/3})$, extending the result of Pach and de Zeeuw (2013) and improving the bound of Charalambides (2012), for the special case where the curve under consideration has a polynomial parameterization. We also derive improved lower bounds for several variants of the sum-product problem in additive combinatorics.

The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and the construction of transcendental numbers. But its usefulness extends beyond the realms of number theory. Other applications of the Subspace Theorem include linear recurrence sequences and finite automata. In fact, these structures are closely related to each other and the construction of transcendental numbers. The Subspace Theorem also has a number of remarkable combinatorial applications. The purpose of this paper is to give a survey of some of these applications including sum-product estimates and bounds on unit distances. The presentation will be from the point of view of a discrete mathematician. We will state a number of variants of the Subspace Theorem below but we will not prove any of them as the proofs are beyond the scope of this work. However we will give a proof of a simplified special case of the Subspace Theorem which is still very useful for many problems in discrete mathematics.

The first open case of the Brown, Erd\H{o}s, S\'os conjecture is equivalent to the following; For every $c>0$ there is a threshold $n_0$ so that if a quasigroup has order $n\geq n_0$ then for every subset of triples of the form $(a,b,ab),$ denoted by $S,$ if $|S|\geq cn^2$ then there is a seven-element subset of the quasigroup which spans at least four triples of the selected subset $S.$ In this paper we prove the conjecture for finite groups.

Let $p_1,p_2,p_3$ be three non-collinear points in the plane, and let $P$ be a set of $n$ other points in the plane. We show that the number of distinct distances between $p_1,p_2,p_3$ and the points of $P$ is $\Omega(n^{6/11})$, improving the lower bound $\Omega(n^{0.502})$ of Elekes and Szab\'o \cite{ESz} (and considerably simplifying the analysis).

Let P_1 and P_2 be two sets of points in the plane, so that P_1 is contained in a line L_1, P_2 is contained in a line L_2, and L_1 and L_2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of P_1xP_2 is \Omega(\min{|P_1|^{2/3}|P_2|^{2/3},|P_1|^2, |P_2|^2}). In particular, if |P_1|=|P_2|=m, then the number of these distinct distances is \Omega(m^{4/3}), improving upon the previous bound \Omega(m^{5/4}) of Elekes.

Let $F(x,y)$ be a polynomial over the rationals. We show that if $F$ is not an expander (over the rationals) then it has a special multiplicative or additive form. For example if $F$ is a homogeneous non-expander polynomial then $F(x,y)=c(x+ay)^\alpha$ or $F(x,y)=c(xy)^\alpha .$ This is an extension of an earlier result of Elekes and R\'onyai who described the structure of two-variate polynomials which are not expanders over the reals.